time-dependent mechanics of living cells
TRANSCRIPT
London Centre for Nanotechnology
Department of Mechanical Engineering
Time-Dependent Mechanics of
Living Cells
Emadaldin Moeendarbary
Dissertation submitted for the degree of
Doctor of Philosophy
University College London
September 2012
Declaration
I, Emadaldin Moeendarbary confirm that the work presented in this thesis is my own.
Where information has been derived from other sources, I confirm that this has been
indicated in the thesis.
.یسپاس گزارم که به من نیروی اندیشیدن داد ، پروردگارا
:به تقدیم
یاری کرده اند ، مرا در دانش آموختنواره م پدر و مادرم که ه
و
.همسرم ژانت که همراه و پشتیبان من در به پایان رسانیدن این کار بود
IV
Acknowledgements
I am sincerely grateful to my main supervisor Dr. Guillaume Charras and second advisor
Dr. Eleanor Stride for their kind guidance and continuous support over the past three
years. I would particularly like to thank Dr. Charras for offering me this project,
welcoming me to carry out this research in his lab and introducing me to biological
techniques. I would also like to extend my grateful appreciation to Prof. L. Mahadevan
for providing great insights on poroelasticity and his valuable inputs to this work.
I appreciate very much the helps of Dr. Marco Fritzsche, Dr. Andrew Harris, Dr. Carl
Leung, and Dr. Richard Thorogate from London Centre for Nanotechnology. I gratefully
acknowledge support of Prof. Nicos Ladommatos and Dr. William Suen from Department
of Mechanical Engineering. I am thankful to our collaborators Prof. Adrian Thrasher and
Dr. Dale Moulding from Institute of Child Health.
I would like to acknowledge the financial support from the Engineering and Physical
Sciences Research Council (EPSRC) through the prestigious Dorothy Hodgkins
Postgraduate Award.
Thank you my friends, Marco, Alireza, Andrew, Cesar and Miia. You are always so
helpful and encouraging.
Thank you, Judy and Guy Whitmarsh for your kind supports.
And many many thanks to my teachers and professors from elementary, junior and high
school to university, who encouraged me to continuously learn and provided me with the
foundations for my academic career.
V
Abstract
Cells sense and generate both internal and external forces. They resist and transmit these
forces to the cell interior or to other cells. Moreover a variety of cellular responses are
excited and influenced by transducing mechanical stimulations into chemical signals that
lead to changes in cellular behaviour. The cytoplasm represents the largest part of the cell
by volume and hence its rheology sets the maximum rate at which any cellular shape
change can occur.
To date, the cytoplasm has generally been modelled as a single-phase viscoelastic
material; however, recent experimental evidence suggests that its rheology can be
described more effectively using a poroelastic formulation in which the cytoplasm is
considered to be a biphasic system constituted of a porous elastic solid meshwork
(cytoskeleton, organelles, macromolecules) bathing in an interstitial fluid (cytosol). In
this framework, a single parameter, the poroelastic diffusion constant pD , sets cellular
rheology scaling as 2~ /pD Ex m with E the elastic modulus, x the hydraulic pore size,
and m the cytosolic viscosity. Though this poroelastic view of the cell is a conceptually
attractive model, direct supporting evidence has been lacking. In this work, such evidence
is presented and the concept of a poroelastic cell is validated to explain cellular rheology
at physiologically relevant time-scales.
In this work, the functional form of stress relaxation in response to rapid application of a
localised force by atomic force microscopy microindentation is examined in detail and it
is shown that at short time-scales cellular relaxations are poroelastic. Then, pD is
measured in cells by fitting experimental stress relaxation curves to the theoretical model.
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Next, using indentation tests in conjunction with osmotic perturbations, the validity of the
predicted scaling of pD with pore size is qualitatively verified. Using chemical and
genetic perturbations, it is shown that cytoplasmic rheology depends strongly on the
integrity of the actin cytoskeleton but not on microtubules or intermediate filaments.
Finally, comparison of scaling of viscoelastic and poroelastic models suggests that short-
time scale viscoelasticity might be due to water redistribution within the cytoplasm and a
simple scaling relating cytoplasmic viscosity to cellular microstructure is provided.
7
Contents
DECLARATION ............................................................................................................................ II
ACKNOWLEDGEMENTS ..........................................................................................................IV
ABSTRACT .................................................................................................................................... V
CONTENTS .................................................................................................................................... 7
LIST OF FIGURES ...................................................................................................................... 12
LIST OF TABLES ........................................................................................................................ 15
SUPPLEMENTARY VIDEO ...................................................................................................... 15
CHAPTER 1 INTRODUCTION ........................................................................................... 16
1.1 A Microscopic view: biological structure of cell ................................................................ 16
1.2 The role of mechanics in cell function .............................................................................. 24
1.3 Experimental techniques for measuring cellular mechanical properties .......................... 27
1.3.1 Atomic force microscopy ............................................................................................. 29
1.4 Cell mechanics and rheology ............................................................................................ 32
1.4.1 The origin of cellular mechanical and rheological properties ...................................... 32
1.4.2 Cell as a continuum media ........................................................................................... 32
1.4.3 Universality in cell mechanics ...................................................................................... 34
8
1.4.4 Top-down approaches to studying cell rheology ......................................................... 36
1.4.5 A bottom up approach: The cell as a dynamic network of polymers ........................... 39
1.4.6 Biphasic models of cells: a coarse-grained bottom up approach ................................ 43
1.5 Aims and motivation ........................................................................................................ 44
1.6 Outline of the following chapters..................................................................................... 45
CHAPTER 2 THEORETICAL OVERVIEW OF CONTINUUM MECHANICAL
THEORIES FOR LIVING CELLS .............................................................................................. 47
2.1 Theory of linear isotropic elasticity .................................................................................. 47
2.1.1 Hertzian contact mechanics: Indentation of elastic material ...................................... 48
2.2 Linear viscoelasticity ........................................................................................................ 51
2.2.1 Storage and loss moduli ............................................................................................... 52
2.2.2 Standard linear solid model ......................................................................................... 53
2.2.3 Power law models ........................................................................................................ 54
2.3 Poroelasticity ................................................................................................................... 56
2.3.1 Theory of linear isotropic poroelasticity ...................................................................... 56
2.3.2 Diffusion equation ........................................................................................................ 57
2.3.3 Scaling law between poroelastic properties and microstructural parameters ............ 58
2.3.4 Poroelastic swelling/shrinking of gels under osmotic stress ........................................ 59
2.4 Basic 1D solutions for fundamental poroelasticity problems ........................................... 61
2.4.1 Uniaxial strain (confined compression) ........................................................................ 61
2.4.2 One dimensional consolidation .................................................................................... 62
2.5 Indentation of a poroelastic material ............................................................................... 67
2.5.1 Indentation of a thin layer of poroelastic material ...................................................... 69
CHAPTER 3 MATERIALS AND METHODS .................................................................... 71
3.1 Cell culture, generation of cell lines, transduction, and molecular biology ....................... 71
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3.2 Fluorescence and confocal imaging .................................................................................. 72
3.3 Cell volume measurements .............................................................................................. 73
3.4 Disrupting the cytoskeleton ............................................................................................. 75
3.4.1 Pharmacological treatments ........................................................................................ 75
3.4.2 Genetic treatments ...................................................................................................... 75
3.5 Visualising cytoplasmic F-actin ......................................................................................... 76
3.6 Atomic force microscopy indentation experiments .......................................................... 78
3.6.1 Measuring indentation depth and cellular elastic modulus......................................... 79
3.6.2 Measurement of the poroelastic diffusion coefficient ................................................ 81
3.6.3 Determining the apparent cellular viscoelasticity ........................................................ 82
3.6.4 Measuring cell height ................................................................................................... 83
3.6.5 Spatial probing of cells ................................................................................................. 84
3.7 Microinjection and imaging of quantum dots .................................................................. 85
3.8 Rescaling of force relaxation curves ................................................................................. 86
3.9 Polyacrylamide hydrogels ................................................................................................ 87
3.10 Particle tracking using defocused images of fluorescent beads ................................... 88
3.10.1 Experimental protocol and calibration setup .......................................................... 88
3.10.2 Radial projection ..................................................................................................... 90
3.11 FRAP experiments ....................................................................................................... 91
3.11.1 Effective bleach radius ............................................................................................ 93
3.11.2 FRAP analysis ........................................................................................................... 93
3.11.3 Estimation of cortical F-actin turnover half-time .................................................... 94
3.12 Data processing, curve fitting and statistical analysis .................................................. 94
CHAPTER 4 THE CYTOPLASM OF LIVING CELLS BEHAVES AS A POROELASTIC
MATERIAL 96
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1.4 Indetation test for characterisation of poroelastic material ........................................ 97
4.1.1 Establishing the experimental conditions to probe poroelasticity .............................. 97
4.1.2 Poroelasticity is the dominant mechanism of force-relaxation in hydrogels ............... 99
4.1.3 Cellular force-relaxation at short timescales is poroelastic ....................................... 102
4.2 Kinetics of whole cell swelling/shrinking........................................................................ 106
4.2.1 Swelling/shrinkage of a constrained poroelastic material ......................................... 108
4.2.2 Poroelasticity can explain the kinetics of cellular swelling/shrinking ........................ 111
4.3 Discussion and conclusions ............................................................................................ 111
CHAPTER 5 POROELASTIC PROPERTIES OF CYTOPLASM: OSMOTIC
PERTURBATIONS AND THE ROLE OF CYTOSKELETON .......................................... 120
5.1 Determination of cellular elastic, viscoelastic and poroelastic parameters from AFM
measurements ........................................................................................................................ 121
5.2 Poroelasticity can predict changes in stress relaxation in response to volume changes . 123
5.3 Changes in cell volume result in changes in cytoplasmic pore size. ................................ 127
5.3.1 Hyperosmotic shock halts movement of quantum dots inside cytoplasm ................ 128
5.3.2 Translational diffusion of fluorescent probes affected by osmotic perturbations .... 129
5.4 Estimation of the hydraulic pore size from experimental measurements of Dp. ............. 133
5.5 Spatial variations in poroelastic properties .................................................................... 133
5.6 Poroelastic properties are influenced by cytoskeletal integrity ...................................... 135
5.6.1 Chemical treatments .................................................................................................. 135
5.6.2 Genetic modification .................................................................................................. 136
5.7 Discussion and conclusions ............................................................................................ 141
CHAPTER 6 GENERAL DISCUSSION AND FUTURE WORK .................................. 148
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6.1 Further discussion .......................................................................................................... 148
6.1.1 Cell rheology and its complexity ............................................................................... 148
6.1.2 Why poroelasticity? ................................................................................................... 149
6.1.3 Direct physiological relevance .................................................................................. 152
6.2 Future work ................................................................................................................... 153
6.2.1 Stress and pressure wave propagation in cells......................................................... 153
6.2.2 Cells as multi-layered poroelastic materials ............................................................. 155
6.3 Conclusions .................................................................................................................... 157
REFERENCES........................................................................................................................... 158
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List of Figures
Figure 1-1 Living cells ...................................................................................................... 17
Figure 1-2 The three major types of filaments that constitute the cytoskeleton. .............. 20
Figure 1-3 Generation of force in the actin-rich cortex moves a cell. .............................. 22
Figure 1-4 Binding proteins induce structural phase transitions to polymerised solution of
purified filaments. ............................................................................................................. 23
Figure 1-5 Cells sense and respond to application of forces. ............................................ 25
Figure 1-6 Common experimental techniques for measurement of cell rheology. ........... 28
Figure 1-7 Atomic force microscopy imaging of the cell surface..................................... 31
Figure 1-8 Universal phenomenological behaviours of the cell and models of cell
rheology. ........................................................................................................................... 34
Figure 1-9 Schematic representation of four different models of cell rheology. .............. 35
Figure 1-10 Dynamic network of filaments determine the cell rheology. ........................ 41
Figure 2-1 The geometry of contact of non-conforming bodies. ...................................... 49
Figure 2-2 The standard linear solid viscoelastic model. .................................................. 55
Figure 2-3 Compression of a porous solid with an interstitial fluid by a porous plunger. 62
Figure 2-4 Creep response of a poroelastic half-space. .................................................... 64
Figure 2-5 Creep response of a finite domain. .................................................................. 65
Figure 2-6 Stress relaxation response of a poroelastic half-space. ................................... 67
Figure 2-7 Spherical indentation and force relaxation a poroelastic material of finite
thickness. ........................................................................................................................... 68
Figure 3-1 Cell volume measurement. .............................................................................. 74
Figure 3-2 AFM experimental setup. ................................................................................ 77
Figure 3-3 AFM force-distance curve. .............................................................................. 80
Figure 3-4 Calculating the height of cell at contact point. ................................................ 83
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Figure 3-5 Spatial AFM measurement .............................................................................. 85
Figure 3-6 High resolution defocusing microscopy of fluorescent bead. ......................... 89
Figure 3-7 Estimation of the effective bleach radius in HeLa cells FRAP experiments... 92
Figure 4-1 Stress relaxation of HeLa cells in response to AFM microindentation ........... 98
Figure 4-2 Force-relaxation curves acquired with different indentation depths on
hydrogel. ......................................................................................................................... 100
Figure 4-3 Normalisation of force-relaxation curves acquired on hydrogel. .................. 101
Figure 4-4 Force-relaxation curves acquired with different indentation depths on cells. 103
Figure 4-5 Normalisation of force-relaxation curves acquired on MDCK cells. ............ 104
Figure 4-6 Normalisation of force-relaxation curves acquired on HeLa cells. ............... 105
Figure 4-7 Experimental setup to investigate the poroelastic behaviour of cells during
swelling/shrinkage. ......................................................................................................... 107
Figure 4-8 Kinetics of cell swelling/shrinking in HeLa cells ......................................... 110
Figure 4-9 Schematic representation of intracellular water movements in different
experimental setups. ........................................................................................................ 118
Figure 5-1 Fitting force-relaxation curves. ..................................................................... 122
Figure 5-2 Poroelastic and elastic properties change in response to changes in cell
volume. ........................................................................................................................... 125
Figure 5-3 Effects of hypoosmotic or hyperosmotic shock on HeLa cells F-actin structural
organization..................................................................................................................... 127
Figure 5-4 Fluorescence recovery after photobleaching in HeLa cells. .......................... 129
Figure 5-5 Changes in cell volume change cytoplasmic pore size. ................................ 132
Figure 5-6 Spatial distribution of cellular rheological properties. .................................. 134
Figure 5-7 Cellular distribution of cytoskeletal fibres. ................................................... 136
Figure 5-8 Ectopic polymerization of F-actin due to CA-WASp. .................................. 137
Figure 5-9 Effect of drug treatments and genetic perturbations on poroelastic properties.
........................................................................................................................................ 138
Figure 5-10 Cell rheology and disease. ........................................................................... 140
Figure 5-11 Effect of volume changes, chemical and genetic treatments on the lumped
pore size. ......................................................................................................................... 143
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Figure 5-12 Schematic representation of the cytoplasm and effects of different cellular
perturbations. .................................................................................................................. 147
Figure 6-1 Microstructural parameters and mesoscopic length scale set the poroelastic
time scale. ....................................................................................................................... 151
Figure 6-2 Stress and pressure wave propagation in cells .............................................. 154
15
List of tables
Table 5-1 Effects of osmotic perturbations on translational diffusion coefficients. ....... 131
Supplementary video
Effects of osmotic shock on the movement of microinjected PEG-passivated quantum
dots in HeLa cell.
Movement of PEG-passivated quantum dots microinjected in a HeLa cell in isoosmotic
conditions (left) and in hyperosmotic conditions (right). Both movies are 120 frames long
totalling 18 s and are single confocal sections. In isoosmotic conditions, quantum dots
move freely (left); whereas in hyperosmotic conditions, quantum dots are immobile
because they are trapped within the cytoplasmic mesh (right). Scale bars = 5 μm.
16
Chapter 1
Introduction
Equation Chapter (Next) Section 1
1.1 A Microscopic view: biological structure of cell
Living cells, the elementary units of life, all carry out similar basic functions and they
display a common architecture despite their various shapes, sizes and internal
complexities (Figure 1-1). In addition to basic functions such as proliferation, protein
synthesis, molecule transport and energy conversion, cells may also perform more
specialised roles that pertain to the organs that they are part of. Prokaryotic cells, such as
bacteria, lack membrane-bounded nuclei while eukaryotic cells contain membrane-
bounded compartments (including nuclei and other organelles) that perform specific
metabolic activities and energy conversion. Plant and animal cells are eukaryotes and are
typically larger (~ 5 to 100 µm) and more complex than prokaryotic cells (~ 1 to 10 µm).
The internal and external environments of the eukaryotic cells are separated by a plasma
membrane which is a selectively-permeable barrier consisting of a lipid bilayer with
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embedded proteins. The work described in this thesis is focused on the animal cell whose
fluid-like plasma membrane cannot alone adopt complex cellular morphologies [1, 2]).
The material inside the cell membrane, excluding the nucleus, is generally referred to as
the cytoplasm which consists of a liquid phase, the cytosol (water, salts, small proteins),
and a solid phase (cytoskeleton, organelles, ribosomes, etc). A general diagram of animal
cell organisation is given in Figure 1-1B.
Figure 1-1 Living cells
(A) Coloured scanning electron micrograph (SEM) of a HeLa cancer cell undergoing cell division. Source:
http://www.sciencephoto.com/media/137830/enlarge. (B) Schematic showing the generic cytoplasm
organisation and nucleus of an animal cell. The main cellular structures and organelles are listed in the right
panel. Source: http://en.wikipedia.org/wiki/File:Biological_cell.svg.
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The cellular skeleton (cytoskeleton) is a complex network of three major types of
polymeric fibre of varying size and rigidity extending throughout the cytoplasm. It
consists of cable-like actin filaments with diameters of ~ 7 nm, rope-like intermediate
filaments with diameters of ~ 10 nm and pipe-like microtubules with diameters of ~ 20
nm [1-3] (see Figure 1-2). The cytoskeleton is responsible for maintenance of cell shape,
cell migration, resistance to externally applied mechanical forces, and cell division.
Furthermore it controls and facilitates the transport of intracellular particles and
organization of cell contents.
Actin filaments are known to be the most important polymer network for the mechanics
of the cytoskeleton. They are organized in a variety of structures such as networks of
filaments within the cytoplasm, the cortex under the plasma membrane, the lamellipodia
at the cell edges, the filopodia and contractile stress fibres. The subunit of an actin
filament (F-actin) is a globular actin protein (G-actin) which accounts for about 5-10
percent of the total cellular protein content [4, 5]. A typical actin filament has a
longitudinal elasticity of ~ 2 GPa and a persistence length of ~ 15 µm which is larger than
its typical contour length of ~ 2 μm. Actin filaments have a distinct polarity originating
from the structural asymmetry of G-actin and filaments exhibit a faster polymerisation
rate at one end (plus barbed-end) compared to the other end (minus pointed-end).
αβ tubulin heterodimers organize into long hollow cylinders to form microtubules which
are the second major constituent of the cytoskeleton [4]. Similar to F-actin, microtubules
are polar filaments because of the structural arrangement of their subunits [2].
Microtubules are straight and rigid with turnovers on the order of minutes [6], and they
play a key role in chromosomal distribution during cell division. They typically radiate
randomly from the centrosome (which is a microtubule organizing centre (MTOC) near
the nucleus) towards the periphery of the cell with their (–) ends proximal to the MTOC
and their (+) ends directed away from the MTOC. The tubular structure of microtubules
gives rise to a higher bending stiffness than actin filaments despite their similar effective
longitudinal Young’s moduli [2]. The persistence length of microtubules is ~ 5 mm which
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is ~ 1000 fold larger than their typical length ~ 5 µm and the typical length of a cell: ~
50 µm. Microtubules are known to be the most significant compressive-load bearing
elements of the cytoskeleton [7].
As the most abundant cytoskeletal filaments, intermediate filaments are located both
inside the cytoplasm and the nucleus and form a large and more diverse family of highly
a -helical proteins. Unlike actin filaments and microtubules subunit proteins, these a -
helical proteins integrate into more diverse sequences with greatly varying molecular
weights to form different types of intermediate filaments such as: keratins that are
abundant in epithelial cells, lamins that are involved in the formation of a structure called
the lamina which is located underneath nuclear membrane and helps stabilise the nuclear
envelope, and vimentin filaments that are expressed mainly in mesenchymal cells [3].
Intermediate filaments, typically have a diameter of ~ 10 nm which is in between the
diameter of actin filaments (7 nm) and microtubules (25 nm).They have a persistence
length of ~ 1 µm that is smaller /comparable to their length giving them more flexibility.
Intermediate filaments extend out from around the nucleus throughout the cytoplasm and
it is thought that they are mechanically incorporated with other parts of cytoskeleton
through specialised crosslinking proteins to reinforce the integrity of the cell [8].
Intermediate filaments are more stable than actin filaments and microtubules (for instance
keratin filaments have a turnover of ~ 30 min [9]) and play a major role in cell-cell
attachments in epithelial tissue [10].
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Figure 1-2 The three major types of filaments that constitute the cytoskeleton.
(A,B,C -I): The electron micrographs and lattice structure of filaments: source:[3] (A,B,C –II): Schematic of
spatial distribution of cytoskeleton filaments. (A,B,C –III): Distribution of cytoskeletal constituents in live
HeLa cells. All images are maximum projections of confocal stacks. Green indicates the localisation of the
GFP-tagged construct. The nucleus is shown in blue in all images. (A-III) F-actin was enriched in cell
protrusions but was also present in the cell body. (B-III) GFP-α-tubulin was homogenously localised in the
cell body. A-III & B-III were acquired by L. Valon. (C-III) Intermediate filaments (keratin 18) were
homogenously distributed in the cell body. Scale bars = 10 μm.
21
As briefly described in the previous paragraphs, each cytoskeletal filament is constructed
from specific protein subunits. There exists a continual flux of constituent subunits
onto/from both ends of filament which results in polymerisation/depolymerisation. In the
steady state, polymerisation and depolymerisation are balanced and as a result the
filament maintains a given length, a phenomenon known as treadmilling (see
Figure 1-3C-III for actin filament). Under normal conditions, the rate of this
polymerisation/depolymerisation (turnover) is different for each of the three major types
of cytoskeletal filament. For instance, intermediate filaments have a much slower rate of
turnover (~ tens of minutes) than actin filaments (~ tens of seconds). Many active features
in the cell such as force generation are in part due to these energy consuming turnover
processes. For instance, actin polymerization [11-13] constructs the leading edge of the
cell, a two dimensional sheet-like structure called the lamellipodium, that drives cell
protrusion (see Figure 1-1 and Figure 1-3).
The above three major filament types are the key parts of the cytoskeleton. To build the
cytoskeletal network architecture and define the mechanics of the cell, these primary
filaments interact with one another, cross-link and bind with themselves and connect to
other parts of cytoplasm (plasma/nuclear membrane, organelles, etc) through various
types of linking proteins and molecular motors. Motor proteins convert the energy
released by ATP hydrolysis into mechanical work to dynamically interact with the
cytoskeleton and to facilitate many different cellular functions such as force generation
and locomotion, cell division and intracellular transport. For instance during cell
movement, myosin II motor proteins move along cytoskeletal filaments and generate
contractile forces necessary for contraction of the cell rear (see Figure 1-3D and
Figure 1-4). Crosslinking proteins organize cytoskeletal filaments into entangled and
bundled complex scaffolds. Hence, they help determine the network architecture leading
to different cellular mechanical functions [14]. For example, the crosslinking protein α-
actinin is one of the most abundant acting binding proteins and it crosslinks actin
filaments into bundles to provide contractility in cytoskeletal assemblies such as stress
22
fibres [15] and the contractile ring [16] (Figure 1-4). Other crosslinkers, such as filamin,
organise actin filaments into a crosslinked gel-like network (Figure 1-4).
Figure 1-3 Generation of force in the actin-rich cortex moves a cell.
(A) Scanning electron microscope image of a fish keratocyte. (B) Phase-contrast image of the keratocyte
moving downward with a speed of ~ 15 µm.s-1. (C-I) Distribution of cytoskeletal filaments with actin
fillaments in brown, microtubules in green and intermediate filaments in blue. (C-II) Electron micrograph of
the F-actin network in the lamellipodium. (C-III) Treadmilling: polymerisation/depolymerisation of actin
filaments by reversible addition of actin subunits to the free ends of the filament. One end of the filament
elongates (plus end) while the other end shrinks (minus end). At steady state, addition and loss are balanced.
(D) Schematic diagram of how cells move forward. Actin polymerisation at the leading edge of the cell
extends a protrusion in the direction of movement. The leading edge adheres to the substrate in front and
adhesions at the rear of the cell are detached. The contractile forces generated by the acto-myosin at the cell
rear push the whole cell body forward. Source:[3].
23
Figure 1-4 Binding proteins induce structural phase transitions to polymerised solution of purified filaments.
(A) Addition of binding proteins with different properties such as type, organisation and concentration results
in the formation networks with different microstructural organisations. Adapted from [17]. (B) Effects of
crosslinkers and myosin II thick filaments on the network topology of actin filaments. (B-I) Entangled
network of purified actin filaments. (B-II) Addition of α-actinin crosslinker proteins changes the shape and
architecture of the actin filament network. (C) Addition of myosin II ,motor proteins leads to further
contraction and rearrangement of the crosslinked F-actin network. Source: [18].
24
1.2 The role of mechanics in cell function
Living cells sense their environment and generate internal and external forces [19]. They
resist and transmit these forces to the cell interior or other cells through chemical and
physical signals. Complex sensory machineries located on different cellular sites such as
primary cilia, stretch-gated ion channels and focal adhesions are responsible for sensing
mechanical stimulations (arising from application of different types of forces such as
tensional, compressional, pressure and shear forces) generated internally or applied
externally on different parts of the cell and eventually transducing these cues into
biochemical signals (see Figure 1-5A) [20, 21]. More fascinatingly, in a closed-loop
feedback manner these mechanosensed biochemical signals trigger energy-consuming
chemical processes relying on ATP hydrolysis that cells employ to modify their own
behaviour and generate forces to respond to and resist the initial mechanical cues.
Mechanotransductory processes greatly influence and control a variety of cellular
responses that lead to changes in cellular behaviour and function such as changes in cell
morphology, lineage and fate [22, 23]. For example, in the musculoskeletal system, actin
and myosin molecular motors produce active contraction in muscle cells in response to
external electrical stimuli such as nerve influxes. Inner hair cells transduce the fluid
oscillation generated in the cochlea by sound waves into electrical signals that are
transmitted to the brain. Vascular endothelial cells bear hemodynamic forces, such as
blood flow shear stress and pressure, and convert these applied stresses into intracellular
signals that influence cellular functions such as proliferation, migration, permeability, and
gene expression [24]. More interestingly, fluid shear stresses can modulate endothelial
structure and function (see Figure 1-5B). Indeed endothelial cells, through rearrangement
of their cytoskeletal structures and active remodelling processes, change their shape and
align themselves with the direction of the flow to control the magnitude of shear forces
they are exposed to [24]. One outstanding challenge in cell biology is to understand how
25
mechanical cues can be transduced into biochemical cues and how biochemical changes
in the cell can give rise to changes in cellular mechanics or the forces applied by cells.
Figure 1-5 Cells sense and respond to application of forces.
(A-I) Application of different types of forces on cell such as shear forces induced by fluid flow over the cell,
tensile forces arising from the extracellular matrix (ECM) and internal compressional or tensional forces
induced and borne by the cellular cytoskeleton. (A-II) Diagram of a mechanosensor at the focal adhesions. At
the focal adhesion site, the internal cytoskeletal forces are balanced by external forces distributed on the ECM
(blue). On the external side of the cell membrane, ECM is attached to focal adhesions (multicolored array
of proteins) through integrins (brown). On the internal side of the cell, membrane actin microfilaments (red)
are anchored into focal adhesions. (III) The internal and external forces are passed through the
mechanosensory site where forces can be sensed, controlled and regulated in a closed-loop feedback. Source:
[20] (B) Endothelial cell morphology transformed by longterm exposure to fluid shear stress. Increasing the
level of shear stress from 0 to > 1.5 Pa causes alignment of bovine aortic endothelial cells (shown under static
culture conditions in I) in the direction of fluid flow after 24 hours (shown in II).Source: [25].
Entanglement (physical) and crosslinking (biochemical) of filamentous polymer networks
inside cells maintain cell shape and govern cellular mechanical properties such as
elasticity. Molecular motors and crosslinkers contract the cytoskeletal polymer network
26
and induce pre-stresses. Active mechanical forces generated from chemical reactions
(such as active polymerisation of filaments or molecular motors) enforce morphogenetic
changes such as cell rounding, cytokinesis, cell spreading, or cell movement. During
these gross morphogenetic changes the cytoskeleton provides the force for
morphogenesis but the maximal rate at which shape change can occur is dictated by the
rate at which the whole cell can be deformed. Hence, the dynamic mechanical properties
(rheology) of cytoplasm, which are poorly understood and are the main topic of this
work, play a major role in setting the rate at which any morphogenetic event can occur.
Furthermore, cells detect, react, and adapt to external mechanical stresses by regulation of
microscopic processes that changes their mechanics. However, in the absence of an in
depth understanding of cell rheology, the transduction of external stresses into
intracellular mechanical changes is poorly understood, making the identification of
the physical parameters that are detected biochemically largely speculative.
Although the correlation between the mechanical properties of tissues and diseases
has been recognized quite widely for a long time, the connection between the mechanics
of living cells (at single cell level) and disease is poorly understood. Indeed, the cell
phenotype in health or disease can be affected by the rheology of the cell. One clinical
example suggests that changes in cell rheology can have consequences for the health of
patients: some patients with a low neutrophil count exhibit a constitutively active
mutation in the Wiskott Aldrich syndrome protein (CA-WASp) that results in over-
activation of actin polymerisation through dysregulated activation of the Arp2/3 complex,
leading to delays in several stages of mitosis [26, 27]. This overabundance of cytoplasmic
F-actin increases cellular apparent viscosity resulting in delays in all phases of mitosis
and causing kinetic defects in mitosis (Moulding et al, Blood, 2012). Further to this
example, there are other reports that identified different mechanical properties for
cancerous cells compared to those of normal cells [28-31] which illustrate the importance
of studying cell rheology. The ultimate goal is to combine theoretical, experimental and
27
computational approaches to construct models for a realistic description of the various
types of cellular mechanical behaviours based on their microscopic cytoskeletal
arrangement, something necessary to enable a true physical understanding of the stress
fields present and generated in tissues and developing embryos during physiological
functions.
1.3 Experimental techniques for measuring cellular mechanical
properties
Depending on the length scale of the cellular structure under investigation and the
required spatial and temporal resolution, many experimental techniques have been
utilized to measure the mechanical properties of the different cellular components. In
general, most methods measure the static/dynamic response of the cellular structure to the
application of controlled forces or deformations. Other techniques track the motion of
endogenous or embedded particles of various sizes to study the rheology of the cell. The
important point is that due to the heterogeneity and complexity of the cell and the
cytoskeleton, the submicrometre-scale measurement techniques can lead to considerably
different evaluations of mechanical properties compared to bulk (several micrometre
scale) measurement techniques. One of the main challenges is to find a universal
framework under which the measured macroscopic properties can be interpreted to obtain
realistic information about the dynamics of the microstructure of the cell or vice versa,
i.e. to estimate the bulk rheological properties from the measured microscopic properties
[32].
28
Figure 1-6 Common experimental techniques for measurement of cell rheology.
Several experimental methods such as magnetic twisting cytometry [33, 34], magnetic
tweezers [35], optical tweezers [36], substrate cell stretchers [37, 38], shear flow
rheometry [39, 40] (see Figure 1-6) and atomic force microscopy (AFM) (see the
following paragraph and Figure 1-7) have been utilized to perturb small regions of the
cell or deform an entire cell to investigate cell rheology. An alternative technique is the
use of passive methods, such as traction force microscopy that involves quantification of
cellular traction forces using different detection mechanisms such as micropillar arrays
[41] or substrate-embedded beads [42]. One more recently applied technique is particle-
29
tracking microrheology that tracks motion of embedded or attached tracer particles
present within the system to extract cellular mechanical properties [43]. Depending on the
cell type, the experimental condition and the probing technique diverse ranges of cellular
rheological properties have been reported and various mechanical models (some of which
are described in the following sections) were applied to interpret quantitative
measurements of cell mechanical properties. Further to the need for significant
improvement in mechanical models, the experimental methods also required to be
improved to measure local/global mechanical properties of single cells more accurately.
Some of these important features include: the ability to impose/measure deformations and
forces on micro and nano scales, perform dynamic experiments, integrate with other
experimental tools such as confocal microscopy and maintain cell viability for a
sufficient period of time [4, 44-46].
1.3.1 Atomic force microscopy
In my studies, I utilized the Atomic force microscope (AFM), a very high resolution form
of scanning probe microscope that was invented in 1986 [47] and has emerged rapidly as
a versatile tool to study biological samples over the past two decades [48]. This allowed
me to acquire topographic images of cells (Figure 1-7D), probe rigidity [49] and
characterisation of viscoelastic properties of different cell lines (Moulding et al, Blood,
2012) and extracellular gel matrices (Calvo et al, Nature Cell Biology, under review).
Topographic AFM measurements involve a tip connected to a micro-fabricated cantilever
being scanned over the surface of a sample in a series of horizontal sweeps (see
Figure 1-7A). A laser beam from a solid state diode is reflected off the back of the
cantilever and collected by photodiodes. Perturbations in the movement of the tip due to
surface topography result in changes in bending of the cantilever and consequently of the
reflection path of the laser beam which is sensed via photodiodes. A piezo-electric
ceramic in a feedback loop is used to move the cantilever up and down to maintain a
constant bending of the cantilever. In contact mode, the tip is touching the surface while
30
sweeping across it. As the deflection of the cantilever is kept constant, the surface
topography is reconstructed in 3D from the piezo movement and the planar trajectory of
the cantilever (see Figure 1-7A). The cantilever vertical and lateral deflections provide
information about the sample height changes and can map the surface distribution of
different chemical functionalities [50].
In addition to high resolution imaging of biological structures, AFM has been used
extensively to assess the mechanical properties of biological materials such as elasticity
and viscoelasticity [51-54]. As will be explained later in Chapter 3.7, AFM indentation
was employed in this thesis to characterise the rheological properties of cytoplasm.
Furthermore AFM can be integrated with other techniques such as defocusing microscopy
to investigate stress propagation inside cells and this idea will be described in the last
chapter.
31
Figure 1-7 Atomic force microscopy imaging of the cell surface.
(A) Schematic diagram of an AFM cantilever tip scanning a surface. Vertical cantilever motion is controlled
by a piezoelectric ceramic through a feedback loop. Bending of the cantilever alters the laser beam reflection
path which is sensed by photodiodes. (B-I,II) SEM images of AFM cantilevers with pyramidal (I) and glued
spherical (I) tips. In (II) a glass microbead was glued onto the cantilever. (C) Phase contrast image of an AFM
cantilever probing HeLa cells cultured on a glass coverslip. (D-I) Contact mode image of HeLa cells shown
in (C), imaged in cell culture medium with a pyramid tipped cantilever. (D-II) Three dimensional
reconstruction of cell topography. Scale bars = 10 µm.
32
1.4 Cell mechanics and rheology
1.4.1 The origin of cellular mechanical and rheological properties
As a first step to studying cellular mechanical and rheological properties, the cell can be
simply considered as a conventional engineering material [55]. Simple conventional
materials exhibit very simple frequency-dependent responses: for normal solids the shear
modulus is independent of frequency and for liquids it is simply proportional to
frequency. However, materials (including cells) with more complex hierarchical
structures and a high degree of heterogeneity can exhibit a variety of complex responses
depending on strength, frequency and spatial application of deformations and forces.
Being such a complex material, the cell displays viscoelasticity, i.e. it behaves like an
elastic solid over short time scales with a finite shear modulus whereas it acts as a viscous
liquid at long time scales [56, 57]. In intermediate regimes, both the storage and shear
moduli govern the mechanical response of the cell. At different hierarchical levels,
individual microstructural elements of the cytoskeleton are perturbed by application of
force and deformed at microscopic scales to accommodate macroscopic deformations.
Furthermore, processes such as continuous turnover of cytoskeletal fibres,
association/dissociation of crosslinkers and activity of molecular motors mean that the
cell is an active biological material [56] that exhibits astonishing rheological behaviours
by coupling active and passive biochemical and mechanical processes. Indeed, time-
dependent measurements of cellular properties reveal a spectrum of relaxation times
which are strongly influenced by the size, stability, geometry, and flexibility of cross-
linkers, active motions, and changes in filament structure due to turnover [45].
1.4.2 Cell as a continuum media
Experiments on cells have shown that the cytoskeleton is the main determinant of cellular
mechanical properties. Hence, much effort has concentrated on describing cytoskeletal
rheology. As will be described in the following, wide ranges of top-down and bottom-up
33
theoretical models have been proposed to describe the rheology of the cytoskeleton and
the cellular environment. A robust model is one that can incorporate the minimum
necessary information about internal structure, spatial granularity, heterogeneity and the
active features unique to cells to predict long-wavelength and long-timescale
redistribution of stress [46, 56]. However the search for such a comprehensive model is
still ongoing. Indeed, even a description of the cell as a multi-layered composite (such as
an elastic membrane tightly bound to the subcortical actin gel, a more granular gel-like
inner layer, a more fluid layer in the cell interior and a stiff nucleus) without
incorporating the internal complexity of each layer and active processes still remains a
very complex system to investigate. Indeed, experimentally measured rheological
properties are highly dependent on the size of the probe as well as its connectivity and
interaction with the cellular structures [58]. For example, during microindentation
experiments with a micron-sized bead, the induced deformations on the cell are
significantly larger than the mesh size of the cytoskeletal network such that continuum
viscoelastic models can be safely used without concern for the heterogeneous distribution
of filamentous proteins in the cytoskeleton. Therefore, operating at sufficiently large
length scales, cells can be considered as continuum media in which the contribution of the
microarchitecture can be coarse-grained through constitutive laws. Despite the limited
ability of continuum models to explain complex cytoskeletal structures, to date they are
still the most efficient means of describing experimental observations on the micrometre
scale and to model stress/strain transmission through the cell. These models are
particularly useful when associated with coarse-grained relations giving scaling between
continuum level rheology and cellular microstructure. Considering the cell as a single
phase material, several types of viscoelastic models have been employed to predict the
biomechanical behaviour of cells [44]. On the other hand, multiphasic models such as
biphasic poroelasticity have been applied to investigate the effects of the different
constitutive phases on cell rheology [4, 59].
34
Figure 1-8 Universal phenomenological behaviours of the cell and models of cell rheology.
(A) The frequency response of cells measured with several measurement techniques. Two master curves were
obtained after rescaling the results from different rheological measurements. Adapted from: [58]. (B)
Spontaneous retraction of a single actin stress fibre upon severing with a laser nanoscissor shows existence of
prestress in the cytosketal bundles. Scale bar = 2 µm.Source:[60]. (C) Anomalous diffusion and response of
the cell to stretch. Spontaneous movements of beads attached firmly to the cell (C-I, II) show
intermittent dynamics. Adapted from [61]. (C-III) The mean square displacement (MSD) of a bead anchored
to the cytoskeleton exhibits anomalous diffusion dynamics (subdiffusive at short time intervals
and superdiffusive at longer time intervals). Red curve indicates the MSD of the bead for a non-stretched cell
and other curves show the MSD of the bead in response to a global stretch measured at different waiting
times ( wt ) after stretch cessation. Scale bars = 10 µm.Source: [62].
1.4.3 Universality in cell mechanics
Cell mechanical studies over the years have revealed a rich phenomenological landscape
of rheological behaviours that are dependent upon probe geometry, loading protocol and
loading frequency [58, 63, 64]. More recent mechanical measurements on eukaryotic
cells agree on the presence of four universal cellular phenomenological behaviours [65]:
1) Cell rheology is scale free: plots of the frequency response of many cell types on log-
log scale display the same shape and follow a weak power law spanning several decades
35
of frequency [58]. 2) Cells are prestressed: mechanical stresses generated continuously by
the internal activity of actomyosin or applied externally on the cell are
counterbalanced by the tensional/compressional state of the cytoskeleton [66]. 3) The
fluctuation dissipation theorem (FDT) breaks down and diffusion is anomalous [61]: The
spontaneous motion of endogenous particles or embedded/attached beads present within
the system does not follow the Stokes-Einstein relationship (see section 1.4.4.2 for
explanation of this relationship). 4) Stiffness and dissipation are altered by stretch:
Application of stretch significantly perturbs the rheological properties of the cell and
depending on the experimental condition the cell can exhibit different behaviours such as
stress stiffening, fluidisation and rejuvenation [62]. See Figure 1-8 for brief illustration of
these universal behaviours.
Figure 1-9 Schematic representation of four different models of cell rheology.
(A) Examples of linear viscoelastic models. (B) Tensegrity [66] (C) Soft glassy rheology [67]. (D) Network
of semiflexible filaments models (with or without effects of crosslinkers and molecular motors).
36
Several top-down (linear viscoelasticity, tensegrity, soft glassy rheology) and bottom-
up (networks of semiflexible polymers) theoretical models (see Figure 1-9) have been
successfully applied to explain the observed phenomenological universal behaviours [64].
However there is still no unifying theory to explain the physical mechanisms that govern
these observed universal cellular behaviours. As will be described briefly in the following
sections, linear viscoelasticity, immobilized colloids/ soft glasses and the network of
semiflexible polymers are the most widely used models that have been proposed to study
cell mechanics.
1.4.4 Top-down approaches to studying cell rheology
1.4.4.1 Linear viscoelastic models
Elastic and viscoelastic constitutive laws have been applied to study the response of the
whole cell to external forces and to investigate the stress and strain distribution inside the
cytoplasm. Whereas the static mechanical properties of cells (elasticity) have been
studied in depth, the time-dependent mechanical properties of cells have received
significantly less attention. The majority of the work to date utilises an empirical
viscoelastic description of cells that assumes the cytoplasm is a single phase homogenous
material [32, 53, 68, 69]. The fundamental properties of simple elastic and viscoelastic
materials are explained briefly in Chapter 2. The most commonly used viscoelastic
models consider the whole cell or part of it as a homogeneous continuum, where the
smallest length scale is significantly larger than the dimensions of the microstructural
constituents. Examples of such models are cortical shell-liquid core, elastic/viscoelastic
solid, and power-law models [44, 64]. The cortical shell-liquid core and
elastic/viscoelastic solid models can generate good fits to experimental data by
introducing a finite number of elastic and viscous elements (springs and
dashpots) coupled in series or parallel leading to exponential decay functions with a finite
number of relaxation times [44]. One example of such a model used to fit stress
37
relaxation is illustrated in Chapter 2.2.2. In contrast to spring-dashpot models, power law
models do not have any characteristic relaxation time and cannot be easily described
using mechanical analogs. Power law structural damping models have been applied more
widely in recent microrheological experiments that measure the viscoelastic response of
the cells over a broader range of frequencies and times. Despite their widespread use, the
major limitation of these models is that they are not mechanistic, fail to relate the
measured rheological properties to structural or biological parameters within the cell, and
thus cannot predict changes in rheology due to microstructural changes.
1.4.4.2 Cell as a soft glassy material: macromolecular crowding and anomalous
diffusion
A large body of more recent research has found that some of the cellular rheological
behaviours are empirically similar to the rheology of soft materials such as foams,
emulsions, pastes, and slurries. Following some experimental observations, it was
proposed that cells could be considered as soft glassy materials [67]. Indeed, measuring
the fluctuations of particles within the cytoplasm revealed that in many cases they exhibit
a larger random amplitude of fluctuations than expected and a greater degree of
directionality than could be developed solely from thermal fluctuations [70]. In a normal
diffusion process (Brownian motion), the mean squared displacement (MSD) of a particle
with size a evolves linearly in time t , 2 6 Tr D t , where TD is the particle
translational diffusion coefficient ( TD can be calculated using the Stokes-Einstein
relationship /6T BD k T apm where m is the viscosity of the medium). However, a
semi-empirical equation 2 ~r ta can be used to describe particle fluctuations when it
exhibits non-Brownian behaviour as is frequently observed in cellular environments [71,
72]. This deviation from the Stokes-Einstein relationship has been attributed to reaction
forces from active cytosketal processes [73] and crowding of a large number of
macromolecules inside the cytoplasm such as mobile intracellular globular proteins and
38
other fixed obstacles like cytoskeletal filaments and organelles [74, 75] that reduce the
available solvent volume and provide barriers to particle Brownian motion [76-78].
Based on these observations, it was proposed that the high concentration of different
proteins in the cytoplasm can lead to liquid crystal and colloidal behaviours that can be
interpreted in terms of the ‘‘soft glassy rheology’’(SGR) model [67]. Crowded colloidal
suspensions or soft glasses exhibit a weak power-law rheology corresponding to a
continuous spectrum of relaxation times. Similar to this, the dynamic modulus of cells
scales with frequency as a weak power law valid over a wide spectrum of time [33, 79].
This semi-empirical SGR behaviour is in contrast to the rheology of semiflexible network
of filaments that generally predict a frequency independent dynamic modulus at low
frequencies [58].
As a conceptual model, SGR explains how macroscopic rheological responses are linked
to localized structural rearrangements originating from structural disorder and
metastability. Briefly, the SGR system consists of particles that are trapped in energy
landscapes arising from their interactions with surrounding neighbours if there exists
sufficient crowding. In such a system, thermal energy is not sufficient to drive structural
rearrangement and as a consequence out of equilibrium trapping occurs. With time
remodelling/rearrangement (micro-reconfiguration) happens when particles escape the
energy barriers of their neighbours and jump from one metastable state to another
reaching a more stable state with relaxation rates slower than any exponential process
[61]. In such a system, injecting agitational energy sourced from non-thermal origins
(such as mechanical shear or ATP-dependent conformational changes of proteins [62])
liberates particles from the energy cages in which they are trapped and facilitates
structural rearrangements causing the material to flow.
39
1.4.5 A bottom up approach: The cell as a dynamic network of polymers
At very short time scales (high frequencies), the cellular viscoelastic dynamic modulus
scales with frequency with a universal exponent of 0.75 [80]. This behaviour is analogous
to the observed rheology of semiflexible polymer networks [81-84]. This analogy as well
as the physical picture of the cell constituted of cytosketal filaments (Figure 1-10A)
suggest that the viscoelastic cellular properties originate mainly from the entropic and
enthalpic interactions between cytoskeletal structures which largely depend on the
architecture and mechanical properties of the constituent cytoskeletal filaments [85].
The flexibility of polymers can be characterised by two length scales, the persistence
length pl and the contour length L [86]. The persistence length or the length of thermal
flexibility pl is the length scale over which thermal bending fluctuations become
appreciable and can change the direction of the filament, /p f f Bl E I k T where
f fE Ik is the bending modulus of a single filament (or flexural rigidity defined by
single filament elastic modulus fE and its cross-sectional second moment of inertia fI )
and Bk T the thermal energy. A filament is called rigid when pL l and flexible when
pL l (Figure 1-10B). However, most cytoskeletal gels are networks of semiflexible
biopolymer chains with the persistence length of individual chains comparable to their
contour length ~ pL l . In such networks, the elastic and dissipative properties originate
from entropic (where energy is stored entropically due to the reduction of the number of
accessible chain spatial configurations in the network [87]) and enthalpic (where energy
is stored in extensional and bending modes of filaments) processes that involve filaments
and also the interaction of filaments with the viscous fluid that they are bathed in [86, 88-
90]. Furthermore, crosslinking of these semiflexible filaments by specialised (rigid or
flexible) proteins at very short distances cl along the length of each individual filament
significantly perturbs the network rheology. Indeed, depending on the properties of
40
crosslinkers such as length, flexibility and concentration different nonlinear elastic effects
can be observed [64].
For purified gels of cytoskeletal filaments such as actin gels, based on a microscopic
picture of a network of entangled and crosslinked semiflexible filaments, various
mechanical models have introduced different scaling relationships relating viscoelastic
properties to microstructural parameters. The persistence length pl , geometric mesh size
l and a characteristic length chL have been shown to determine the viscoelasticity of
these purified gels. For an actin gel with an actin concentration Ac and a monomeric actin
size a , the geometrical mesh size (a measure of the nearest distance between two
neighbouring monomers forming part of different filaments) is given by ~1/ Aacl
[91]. In dense network regimes ( pa ll ) the elastic modulus scales as
2 2 3~ B p eE k T l ll where el is the entanglement length (the confining length scale that
restrict topological motion of neighbouring filaments). In this entangled network the
characteristic length ch eL l is the entanglement length el and the physical effect of
crosslinking from entanglements is to constrain the motion of filaments to a tube-like
region, with a diameter 3/2 1/2~e e pd l l , surrounded by other filaments (Figure 1-10C-I).
On the other hand for a densely crosslinked gel, the mesh size ( scales the same as the
entanglement length) is the characteristic length of the network ~ch eL l l
(Figure 1-10C-II) and the elastic modulus scales as 5 2~ B pE k T ll .
41
Figure 1-10 Dynamic network of filaments determine the cell rheology.
(A) Image of actin filaments stained with Alexa Fluor 647 phalloidin in a COS-7 cell taken by dual stochastic
optical reconstruction microscopy (STORM). The z-position from the substrate is colour coded with respect
to the scaling shown in the colour scale bar. Scale bars = 2 μm. Source: [92]. (B) The contour length L of a
polymer compared to its persistence length pl changes the dynamics of polymer networks from flexible to
semiflexible and rigid regimes (from left to right respectively). (C) Schematic representation of minimal
models for the dynamics of networks of polymers. Semiflexible polymer chains such as actin filaments
exhibit complex dynamics under steric entanglements or crosslinks. (C-I) Tube picture of polymer dynamics.
Two length scales, l the mesh size and el the entanglement length (or ed the tube diameter), set the network
dynamics. (C-II) For densely crosslinked polymers the mesh size l or entanglement length ~el l sets the
dynamics of network. Source: [32].
42
The experimentally determined elastic moduli of cells were found to be several orders of
magnitude larger than those measured in in vitro studies of stress-free F-actin gels and
this suggested that the main elastic properties of cells could not result solely from
semiflexible networks in a stress free state, such as networks of filamentous actin [93].
This was explained by the fact that unlike most conventional materials, the viscoelastic
response of semiflexible networks of biopolymers is highly nonlinear (tension or
deformation-dependent) and thus the prestressed state of filaments in the cytoskeleton
could result in the high measured elasticities for cells [94]. In addition to the external
application of tensional forces, active intracellular processes such as contraction mediated
by myosin motor proteins could also generate this prestress. The prestress is transmitted
through the cytoplasm by the actin cytoskeleton and balanced by adjacent cells and the
extracellular matrix. Furthermore as suggested in the tensegrity model [66, 95], the
prestress distribution inside the cytoplasm might be partly balanced by compression of
other cytoskeletal filaments such as microtubules.
Considering the cytoskeletal network as a physical gel of purified cytoskeletal filaments
is a first step for studying mechanical properties of the cell but yet too simplistic.
Indeed cytoskeletal filaments and proteins continuously consume energy released from
ATP hydrolysis that turns the system into a non-equilibrium state. This ATP energy is
involved in filament polymerisation/depolymerisation, active association/dissociation of
crosslinkers and the activity of molecular motors. New theoretical models have been
recently proposed to study such non-equilibrium systems [57, 96]. One category of
models considers the detailed dynamics of filaments and their interactions with molecular
motors and builds up mechanical properties of the filamentous system using microscopic
equations and numerical simulations [97]. Alternatively hydrodynamic models describe
the behaviour of the mixture by establishing phenomenological relations between a few
coarse-grained variables [96, 98].
43
1.4.6 Biphasic models of cells: a coarse-grained bottom up approach
Considering a cell to be a single phase material is counterintuitive given that more than
60 percent of the cellular content is water. In mammalian cells, most studies view water
solely as a solvent and an adaptive component of the cell that engages in a wide range of
biomolecular interactions [99]. However less attention has been given to the significance
of water in the dynamics of the cytoskeleton, its role in cellular morphology, and motility.
Recent experimental work showed the presence of transient pressure gradients inside cells
and suggested that these could be explained by the biphasic nature of cytoplasm [100-
103]. As a consequence, a biphasic description of cells was proposed based on
poroelasticity (or biphasic theory), in which the cytoplasm is biphasic consisting of a
porous elastic solid meshwork (cytoskeleton, organelles, macromolecules) bathed in an
interstitial fluid (cytosol) [100, 101, 103-105]. In this framework, the viscoelastic
properties of the cell are a manifestation of the time-scale needed for redistribution of
intracellular fluids in response to applied mechanical stresses and the response of the cell
to force application depends on a single experimental parameter: the poroelastic diffusion
constant pD [106], with larger poroelastic diffusion constants corresponding to more
rapid stress relaxations. A minimal scaling law 2~ /pD Ex m relates the diffusion
constant to the drained elastic modulus of the solid matrix E , the pore size of the solid
matrix x , and the viscosity of the cytosol m [100]. Therefore, contrary to viscoelastic
models, the dynamics of cellular deformation in response to stress derived from
poroelasticity can be described using measurable cellular parameters, allowing
changes of rheology with E , x , and m to be predicted which makes this framework
particularly appealing conceptually.
44
1.5 Aims and motivation
One of the most striking features of eukaryotic cells is their capacity to change shape in
response to environmental or intrinsic cues driven in large part by their actomyosin
cytoskeleton. In studies of cellular morphogenesis, the cytoplasm is generally viewed as
an innocuous backdrop enabling diffusion of signalling proteins. This view fails to
account for the fact that the time-dependent mechanical properties (or rheology) of the
cell are determined by the cytoplasm because it forms the largest part of the cell by
volume. During gross morphogenetic changes such as cell rounding, cytokinesis, cell
spreading, or cell movement, the cytoskeleton provides the force for morphogenesis but
the maximal rate at which shape change can occur is dictated by the rate at which the
cytoplasm can be deformed. Using a combination of techniques from molecular cell
biology and nanotechnology, my aim is to investigate the biphasic nature of animal cells
and its implications for cell rheology. I examine the contribution of intracellular water
redistribution to cellular rheology at time-scales relevant to cell physiology (0.5-10s) and
investigate how the cytoskeleton and macromolecular crowding interact to set cellular
rheology.
Experimental measurements of cellular viscosity are generally interpreted with models
that either consider the cell as a single phase homogenous material or describe cell
rheology with a continuous spectrum of relaxation times. These models lack predictive
power because rheological parameters cannot be related to cellular structural and
constitutive parameters. Recent studies [100-102, 107] suggest that water plays an
important role in cellular mechanical properties. Based on the physical nature of
cytoplasm, it was suggested that cells are like fluid-filled sponges. One very interesting
implication of the fluid-sponge model is that cellular rheology depends on only one
parameter, the poroelastic diffusion constant, that scales with three structural and
constitutive parameters: the sponge elasticity, the sponge pore size, and the viscosity of
45
the interstitial fluid. Though this is a conceptually attractive model, direct experimental
evidence to validate this model has been lacking.
In this thesis, such evidence is presented and I seek to study cell rheology within the
framework of poroelasticity. I ask whether cell rheology exhibits a poroelastic behaviour
similar to well characterised poroelastic materials such as hydrogels and then I investigate
the validity of a poroelastic scaling law by perturbing cellular parameters that are
important in setting poroelastic properties. To study cell rheology two experimental
approaches were optimised: AFM micro-indentation stress-relaxation tests and whole cell
swelling/shrinking experiments. Cellular stress relaxations were measured by rapidly
applying a localized force onto cells via an AFM cantilever and monitoring the ensuing
stress relaxation over time (Figure 3-2). As an alternative approach, the ability of the
poroelastic theory to predict the swelling/shrinking kinetics of the cells was investigated
by applying sudden osmotic perturbations and monitoring associated dynamics by
defocusing microscopy (Figure 4-7).
1.6 Outline of the following chapters
In Chapter 2, the basic principles of elastic, viscoelastic and poroelastic continuum
mechanical theories are introduced. In particular in relation to indentation experiments,
Hertzian contact mechanics is briefly overviewed. The theory of linear isotropic
poroelasticity is explained more comprehensively followed by the solutions of some
fundamental problems useful in interpreting the presented experimental measurements.
In Chapter 3, the techniques and methodologies employed to determine the poroelastic
properties of cells are presented. These include cell preparation, chemical and genetic
treatments, AFM indentation experiments, particle tracking as well as fluorescent and
brightfield microscopy.
46
In Chapter 4, the ability of the poroelastic theory to describe cellular stress relaxation in
response to rapid application of a localised force by AFM are investigated. The cellular
force-relaxation curves are compared with stress relaxation curves acquired on well
characterised poroelastic hydrogels and the functional form of these curves is discussed in
detail. Also in this chapter, the kinetics of cell swelling/shrinking is studied by employing
poroelasticity to describe the dynamics of cell volume changes induced via sudden
osmotic perturbations and monitored by defocusing microscopy.
In Chapter 5, using indentation tests in conjunction with osmotic perturbations, the
validity of the predicted scaling of pD with pore size is verified qualitatively. Also using
chemical and genetic perturbations, the dependence of cytoplasmic rheology on the
cytoskeleton is explored. In this chapter, the significance and the implications of the
experimental results for understanding the biological determinants of cellular rheology
are discussed.
In Chapter 6, the conclusions from this work are summarised and the implications of
poroelasticity for cell mechanics are discussed further. Also, prospective work and further
experiments that can be performed in the future are presented, including other
experimental techniques to investigate time-dependent mechanical properties of cells.
47
Chapter 2
Theoretical overview of continuum
mechanical theories for living cells
Equation Chapter (Next) Section 1
2.1 Theory of linear isotropic elasticity
The elastic framework is the simplest continuum formulation that provides the
relationship between deformations/strains and forces/stresses. Linear elasticity is the very
simplified version of finite strain theory in which the material strains ( ε strain tensor) are
considered to be infinitesimal (small deformations) and have a linear relationship with
imposed stresses (σ stress tensor). From Newton’s second law, the equation of motion for
such material transforms into
σdiv r rB u (2.1)
where r is the density of material, B body force per unit mass, u is the vector of solid
displacement for small deformations:
48
ε .0.5 Tu u (2.2)
The div and designate the divergence and gradient operators, respectively. For an
isotropic and homogeneous material, the stress-strain relationship is expressed through
the constitutive equations:
σ ε ε 2 tr( )m l I (2.3)
where m and l are Lamé’s constants, I the identity tensor and tr the trace operator. The
first Lame constant m is equivalent to the shear modulus G . The following equations
define the relationships between Lame’s constants , the elastic Young’s modulus E and
the Poisson ratio n [108]
,2(1 )2
.1 2
Em
nmn
ln
(2.4)
Using equations (2.1), (2.2), and (2.3), the stress field can be computed from a prescribed
displacement field and boundary conditions. On the other hand, compatibility conditions
[108] are required to derive a unique displacement field knowing the components of the
stress tensor. Typically in most materials, including cells, the body forces and inertial
terms can be neglected and thus the equation (2.1) transforms into equilibrium equation,
σdiv 0.
2.1.1 Hertzian contact mechanics: Indentation of elastic material
Here the Landau and Lifshitz [109] approach is presented for studying the contact
problem for spherical linear isotropic elastic bodies which is known as Hertz’s contact
problem. It can be shown that when two smooth surfaces contact each other without
deformation (Figure 2-1A) their separation in the z direction h can be expressed in terms
of their principal relative radii of curvature. For the contact of two spheres with radii of
1R and 2R , the separation takes the form:
2 2
2 2*
1 2
( )1 1 1( ) ,
2 2x y
h x yR R R
(2.5)
49
where x and y are coordinates lying on the tangent plane of the contact with the contact
point as the origin and 1 2 1 2* /R RR R R the relative radius of curvature. Now let
us apply a total compressive load of F to the touching spheres such that the centres of
the two spheres approach one another by an amount 1 2d d d . The separation distance
between two bodies at a given height z , can be calculated from geometrical
considerations shown in Figure 2-1B:
'1 2( )h h w wd (2.6)
where 1( , )w x y and 2( , )w x y are the elastic deformations for surface points on each body
respectively. It is trivial that within the contact area ' 0h and outside the contact area
' 0h so that the surfaces do not overlap.
Figure 2-1 The geometry of contact of non-conforming bodies.
(A) Unloaded case where the curved objects are just touching. (B) Loaded case where the curved objects are
pressed into each other. Source: [110].
Considering the Hertzian conditions (i.e. continuous, non-conforming and frictionless
surfaces, small elastic strains where the radius of contact circle a is much smaller than
the relative radius of curvature *a R and that each sphere can be considered as an
50
elastic half-space), the normal displacement of each surface 1( , )w x y or 2( , )w x y , can be
written in terms of the normal component of pressure p on the surface:
' '
' '' 2 ' 2
( , )1( , ) .
2 ( ) ( )
p x yw x y dx dy
x x y y
npm
(2.7)
Within the contact area, ' 0h , substituting the above equation into the geometrical
relationship (2.6) yields:
' '1 2 ' '
' 2 ' 21 2
2 2
1 2
( , )1 1 12 ( ) ( )
1 1 1( ).
2
p x ydx dy
x x y y
x yR R
n np m m
d (2.8)
Solving (2.8) by applying potential theory [109], results in the following relationships for
the pressure field distribution and the contact radius
23
( ) 1 ,2 m
rp r p
a (2.9)
1 2 1 2
1 2 1 2
3 1 18 m
RRa p
R Rp n n
m m (2.10)
where 2 2r x y and mp is the average pressure over the contact area
2
0( )2 .
a
mF p r rdr p ap p (2.11)
Substituting the pressure field ( )p r into equation (2.7), we find the normal displacement:
2 2*
1 3( ) 2 ,
4mpw r a ra
n pm
(2.12)
with the equivalent shear modulus *m defined as:
1 2
*1 2
1 1 1.
n nm m m
(2.13)
Using above the equation and substituting equations (2.5) and (2.12) into equation (2.6)
one can obtain the following relationship within the contact area:
51
2
2 2* *
32
16 2mp ra r
a Rp
dm
(2.14)
from which the mutual approach of the distant points in the two solids can be derived:
*
3.
8mp ap
dm
(2.15)
All parameters can be expressed in terms of total load F as in practice this parameter is
specified:
*
3 * * 3/2*
8 8,
3 3F a R
Rm
m d (2.16)
where *a R d . One special case of the Hertz contact problem which is of interest to
us is where one sphere is rigid ( 1m ) and of finite radius 1R and the other sphere is
elastic 2m but 2R . This is the case of indentation of a semi-infinite domain and for
this case the force-indentation relationship is
2 3/21
2
8.
3(1 )F R
md
n (2.17)
2.2 Linear viscoelasticity
A viscoelastic material undergoing deformation simultaneously stores and dissipates
mechanical energy. Viscoelasticity is the phenomenological theory which describes the
time-dependent response of such material. In this framework regardless of the
microstructure of the material, the relaxation processes are represented by exponential or
power law functions of time. In models based on exponential relaxation processes, stress
or strain functions can be represented by combining spring and dashpot elements. In such
models, the resultant exponential functions can have a single relaxation time if there is
only one dashpot element or a distribution of relaxation times if there are several. On the
other hand, another type of relaxation model that cannot be represented by simple
mechanical analogues but is encountered experimentally widely in the dynamics of
complex materials is power law relaxation.
52
2.2.1 Storage and loss moduli
The stress-strain relationship for a linear viscoelastic isotropic material can be expressed
in the time domain as:
ε ε ε
σd d d
2 ( ) d ( ){tr } d ( ) d ,d d d
t t t
t t G tam t t l t t t tt t t
I (2.18)
where εd /dt is the shear rate and ( )G ta t is the relaxation modulus designating the
deviatoric and the dilational part of the stress-strain relationship by 1a and 2a
respectively [111]. This equation implies that small changes in strain can be integrated
over time to yield the total stress.
The response of a material under oscillatory excitations is a typical way of measuring
shear viscoelastic properties. Let us assume that the material is subjected to an oscillatory
strain with amplitude ε0 and frequency w . Substituting a strain of the form
ε ε0( ) sint tw (with the corresponding strain rate of ε ε0d /dt= cos tw w ) into equation
(2.18) and performing some algebra gives the stress-strain relationship:
σ ε ε
εε ε ε
0 0
0 0
0 0
( ) ( )sin d sin ( )cos d cos
dsin cos ,
dt
t G t G t
GG t G t G
w t wt t w w t wt t w
w ww
(2.19)
where 0
( )sin dG Gw t wt t is the shear storage modulus and
0( )cos dG Gw t wt t the shear loss modulus. The physical meaning of the storage
and loss moduli can be explained as follows: if the material response is purely elastic then
the storage modulus G is the non-vanishing term in equation (2.19). In this case G
appears as the frequency-dependent shear modulus similar to a simple elastic material
where the stress is in phase with the strain: σ ε~ m . On the other hand, for a purely
viscous material 0G and the loss modulus G is the only non-vanishing term. This
results in a stress-response that has a phase lag of /2p with strain and /G w acts as the
53
frequency-dependent dynamic viscosity (similar to Newtonian fluid behavior
σ ε~ d /dth , where h is the viscosity). Unlike purely elastic or purely viscous materials,
viscoelastic materials exhibit a mixed behaviour with a phase lag of j (0 /2j p )
between the applied strain and the resultant stress.
From an experimental point of view, one way to determine the viscoelastic properties of a
material is to measure the stress response of the material in response to an applied
oscillatory strain of the form 0( ) sint te e w . Let us consider the measured stress-
response to have the form 0( ) sint ts s w j where 0s is the amplitude of stress
oscillations. Using equation (2.19), the following relationships can be used to calculate
the frequency-dependent storage and loss moduli of the material:
0
00
0
cos ,
sin .
G
G
sj
es
je
(2.20)
2.2.2 Standard linear solid model
In this section the standard linear solid model (also known as the Zener model) which has
been widely used to describe the time-dependent behaviours of biomaterials such as
living cells [51, 53, 54] (see Figure 2-2A) is presented. The differential equation
describing this model is:
1 1 2 1 2
d d.
dt dtK K K K K
s eh s h e (2.21)
Applying a sudden constant strain 0 ( )H te e (where ( )H t is the Heaviside step
function) or a sudden constant stress 0 ( )H ts s on a Zener viscoelastic material results
in a stress-relaxation or creep response :
1
0 2 1 ,Kt
K K e hs e Figure 2-2B, (2.22)
54
1 2
1 20 1
2 1 2
1 ,K K
tK K
Ke
K K Kh
se Figure 2-2C. (2.23)
Performing some mathematical substitutions, equation (2.21) can be solved to obtain the
dynamic response of a Zener material to an oscillatory load with G and G taking the
form (as plotted in Figure 2-2D):
2 2 21 2 2 1
2 2 21
21
2 2 21
( ),
K K K KG
KK
GK
h wh w
hwh w
(2.24)
2.2.3 Power law models
One of the most surprising features of a wide range of soft materials is that they display a
power law behaviour in which their relaxation spectrum lacks any characteristic time
scale. It is surprising because materials (including living cells) with very different
microstructural organization exhibit the same empirical behaviour. The creep and stress
relaxation responses of such power-law material can be written in the form:
0 0
0
0 0 '0
( ) /
( )
tt K
tt
t Kt
b
b
e s
s e (2.25)
where 0K is the parameter characterizes the softness or compliance of the material and 0t
and '0t are the normalization timescales that can be chosen arbitrary [64]. In other words,
0K is the ratio of stress to unit strain measured at an arbitrary chosen time 0t [112]. The
timescale invariant behavior of power law functions comes from the fact that changing 0t
or '0t does not affect the value of b . This equation can describe linear elastic and
Newtonian viscous behavior for b = 0 and b = 1 respectively. Using this equation, non-
exponential relaxation processes can be described in a very economical way with a small
number of parameters. Considering equations (2.19) and (2.25), the response of the
55
material to dynamic loading also follow a power law with the same exponent b ,
'', ~G G bw [113, 114].
Figure 2-2 The standard linear solid viscoelastic model.
(A) The standard linear solid model consists of a spring 2K in parallel with a spring 1K in series with a
dashpot h . (B) Stress-relaxation: the stress relaxes exponentially over time in response to application of a
sudden constant strain. (C) Creep: in response to application of a sudden constant stress, the strain increases
instantaneously until it reaches its final value. (D) Dynamic response of Zener model (with 1 1K pa,
2 0.1K pa and 1h pa.s): plot of the storage and loss moduli as a function of applied frequency.
56
2.3 Poroelasticity
The theory of poroelasticity studies the behaviour of a porous medium consisting of an
elastic solid matrix infiltrated by an interconnected network of fluid saturated pores.
2.3.1 Theory of linear isotropic poroelasticity
In this section the governing equations for a linear isotropic poroelastic material are
presented [115]. The constitutive equations are an extension of linear elasticity to
poroelastic materials first introduced by Biot [116]. Alternatively one can use the biphasic
model [117] that has been extensively applied in modelling the mechanics of articular
cartilage and other soft hydrated tissues. Biot’s formulation can be simplified when
poroelastic parameters assume their limiting values. Under the “incompressible
constituents” condition, the material exhibits its strongest poroelastic effect and the Biot
poroelastic theory can be mathematically transformed to the biphasic model.
Constitutive law Let us consider the quasistatic deformation of an isotropic fully
saturated poroelastic medium with a constant porosity. The constitutive equation relates
the total stress tensor σ to the infinitesimal strain tensor ε of the solid phase and the pore
fluid pressure p :
σ ε ε -2
2 tr( ) ,(1 2 )
s s
ss
GG p
n
nI I (2.26)
where sG and sn are the shear modulus and the Poisson ratio of the drained network
respectively, and εtr ( )q the variation in fluid content. This equation is similar to the
constitutive governing equation for conventional single phase linear elastic materials.
However the time dependent properties are incorporated through the pressure term that
acts as an additional external force on the solid phase.
Equilibrium equation In the absence of body forces and neglecting the inertial terms,
the local stress balance results in the equilibrium equation
57
σdiv 0. (2.27)
Darcy transport law Darcy’s law considers fluid transport through the non-
deformable porous medium
K pq (2.28)
where q is the filtration velocity and K the hydraulic permeability.
Continuity equation Neglecting any source density, the mass conservation of a fluid
yields:
div .tq
q (2.29)
2.3.2 Diffusion equation
Combining the continuity equation and Darcy’s law results in
2 .K pt
q (2.30)
Applying the equilibrium condition to the constitutive law, one can obtain the Navier
equations
2 div 0,(1 2 )
ss
s
GG p
nuu + (2.31)
where u is the vector of solid displacement for small deformations
ε 0.5 Tu u , and 2 designates the Laplacian operator.
The diffusion equation for εtr ( )q is obtained by combining equations (2.30) and
(2.31)
2PDt
qq (2.32)
where pD is the poroelastic diffusion coefficient:
58
2 (1 )(1 2 )s s
ps
GD K
nn
(2.33)
Equation (2.32) is the fundamental equation in the theory of consolidation which deals
with the time dependent settlement of porous materials. Derivation of the diffusion
equation implies that under the assumptions made in this section, there are three
independent sets of parameters ( sG , sn and K ) that characterize the mechanical
properties of a poroelastic medium.
2.3.3 Scaling law between poroelastic properties and microstructural parameters
One important consequence of considering a poroelastic cytoplasm is that the measured
macroscopic mechanical properties of the cell can be related to some coarse-grained
cellular microstructural parameters and the hydraulic pore size x of the cytoplasm can be
determined as a first step to understanding the microstructure of the cell. Considering a
simple Poiseuille flow inside a tube of radius r , the relationship between the flow rate Q
and the pressure gradient p is given by
4
,8r
Q pp
m (2.34)
where m is the viscosity of the fluid. For a porous body that contains n straight tubes per
unit area with porosity 2n rj p the filtration velocity is
4 2
.8 8n r r
q nQ p pp jm m
(2.35)
Taking into account irregularities, interconnectivities and tortuosities of the pores in a real
porous matrix the above relationship reads:
2
,4r
q pjmk
(2.36)
where k is the Kozney constant [118]. Considering equation (2.36) and a simple analogy
between the described Poiseuille flow and the flow through the porous matrix with an
59
average pore size x (Darcy’s law) leads to the following relationship for the hydraulic
permeability K
2
,4
Kj xk m
(2.37)
Substituting this equivalent expression for the hydraulic permeability into equation (2.33)
results in:
2(1 ).
(1 )(1 2 ) 4s s
ps s
ED
n j xn n k m
(2.38)
As a first approximation all of the parameters inside the parenthesis are assumed to be a
constant a and the functional dependence of all parameters with respect to the porosity
of the structure j is neglected. Therefore a fundamental scaling law for poroelastic
cytoplasm takes the form:
2
~ ,p
ED
xm
(2.39)
where m is interpreted as the interstitial fluid viscosity, and 2 1s sE G n the average
elasticity of the constituent solid network.
2.3.4 Poroelastic swelling/shrinking of gels under osmotic stress
The fundamental poroelastic equations for swelling/shrinking of gels under osmotic
perturbation are very similar to the ones presented in section 2.3.1 with a redefinition of
some parameters such as pore pressure and fluid flux [119]. Let us consider the initial
state of the gel to be homogeneous and free from mechanical load with 0C the initial
concentration of solvent inside the gel and 0m the initial chemical potential of the gel.
Equation (2.29) rewritten in terms of the concentration of the solvent in the gel is:
div .Ct
q (2.40)
Darcy’s law defines the migration of the solvent in the gel, thus equation (2.28)
transforms to
60
2
,K
mq (2.41)
where is the volume per solvent molecule and m the time-dependent chemical
potential of the solvent. Incompressibility of the network and the solvent imply that the
change in volume of the gel is only due to migration of solvent molecules into or out of
the network which changes the volume of solvent inside the gel:
ε 0tr( ) ( ).C C (2.42)
At thermodynamic equilibrium, the work done on each element of the gel equals the
change in free energy Wd written as σ ε 0tr( ) ( )W Cd d m m d. , where W is the
Helmholtz free energy per unit volume of the gel. In this relationship σ ε( . )tr d is the
mechanical work due to stress and 0( ) Cm m d is the work done by the chemical
potential. Using equation (2.42) one can write the change in free energy in terms of only
the strain tensor σ ε ε0tr( ) ( )tr /Wd d m m. which yields:
ε
σε
0( ) ( ).
W m mI (2.43)
The free energy can be written as a function of the strain tensor in the linear case of an
isotropic gel:
ε ε2 2[tr( ) ( ) ].(1 2 )
s
ss
W G trn
n (2.44)
Combining equations (2.43) and (2.44) results in equation (2.26) replacing pore pressure
by the term 0( )/m m :
σ ε ε - 02
2 tr .(1 2 )
s s
ss
GG
n m mn
I I (2.45)
Navier and diffusion equations similar to the Biot poroelasticity equations can then be
derived using the above equations:
21
div 0,(1 2 )
ss
s
GG m
nuu + (2.46)
61
2 .P
CD C
t (2.47)
These equations can be used to define the dynamics of gel swelling/shrinkage in response
to a change in extracellular osmolarity. Experiments in Chapter 4.2 examines this
situation in conjunction with the measurement of surface displacements of the gel by
defocusing microscopy to determine PD .
2.4 Basic 1D solutions for fundamental poroelasticity problems
2.4.1 Uniaxial strain (confined compression)
Let us consider the situation where all the components of the strain tensor except
/xx xu xe are zero so that the poroelastic material is only allowed to deform in one
direction. In this case the non-zero components of the stress tensor reduce to
-
-
2 (1 )
(1 2 )2 (1 2 )
.(1 2 ) (1 ) (1 )
s s
xx xxs
s s s syy zz xx xx
s s s
Gp
Gp p
ns e
nn n n
s s e sn n n
(2.48)
Substituting xu as the only component of the displacement field into the Navier equation
(2.31), reads
2
2
2 (1 )0
(1 2 )s s x
s
G u px x
nn
(2.49)
Combining equations (2.30) and (2.48) and using the fact that xxe q , the diffusion
equation for the pore pressure is obtained:
2
2xx
p
p pD
t x ts
(2.50)
Contrary to equation (2.32), equation (2.50) is an inhomogeneous diffusion equation that
can be solved by specifying a stress condition. In the following the one dimensional time-
62
dependent settlement of a poroelastic material subjected to certain types of boundary
conditions is studied.
2.4.2 One dimensional consolidation
Finite domain creep problem Consider a porous plunger that compresses a poroelastic
material confined in a cylinder of height L as shown in Figure 2-3. At 0t a certain
constant load 0(0, ) ( )xx t H ts s is applied on the solid matrix and held constant. Under
this condition the pressure diffusion equation (2.50) transforms into a homogeneous
diffusion equation
2
2.p
p pD
t x (2.51)
Figure 2-3 Compression of a porous solid with an interstitial fluid by a porous plunger.
The fluid diffuses out through the meshwork and exits through the plunger (red arrows) applying the external
load.
63
Assuming an ideal permeable plunger and zero flux at the bottom of the cylinder yields
0p at 0x and / 0p x at x L respectively. Considering the above
boundary and initial conditions one can solve equation (2.51) analytically:
2 20
1,3,..
4( , ) sin exp
2n
n xp x t n
m Lp
s p tp
(2.52)
where 2/4pD t Lt is a dimensionless time. The solid phase displacement field is
calculated employing equation (2.49) and the following boundary conditions 0xu at
x L :
2 20 2 2
1,3,..
(1 2 ) 8( , ) cos 1 exp
2 (1 ) 2s
xs s n
L n xu x t n
G n Ln p
s p tn p
(2.53)
Semi-infinite domain creep problem Now I consider the case where the poroelastic
material inside the cylinder is infinitely extended in the x direction. In this case, all the
boundary and initial conditions are the same as for the finite domain problem except that
0xu when x which implies that the dissipations arising from solid fluid
interactions vanishes to zero at infinity, / 0p x . The solutions for the pore pressure
and displacement field for a creep problem of a semi-infinite poroelastic domain can be
written as
0( , ) erf2 p
xp x t
D ts (2.54)
2
40
(1 2 )( , ) 2 erfc .
2 (1 ) 2p
xs D t
xs s p
Dt xu x t e x
G D tn
sn p
(2.55)
Figure 2-4 illustrates the normalized pore pressure and displacement field as a function of
depth at different times for 1pD . At 0x The settlement on the surface is
0
(1 2 )(0, ) 2 .
2 (1 )s
xs s
Dtu t
Gn
sn p
(2.56)
64
The solutions for application of a sudden loading of the top surface by a fluid with
pressure 0p can be derived similarly assuming the following boundary conditions
0(0, ) ( )p t p H t and (0, ) 0xx ts . The displacement solutions are exactly similar to
equations (2.53) and (2.55) replacing 0s with 0p . However the pore pressure solutions
take the form
2 20
1,3,..
4( , ) 1 sin exp ,
2n
n xp x t p n
m Lp
p tp
(2.57)
0( , )2 p
xp x t p erfc
D t (2.58)
for finite and infinite domains respectively.
Figure 2-4 Creep response of a poroelastic half-space.
Pore pressure (left) and displacement (right) solutions for the semi-infinite domain creep problem. The curves
are plotted as a function of depth at different times for 1pD .
In our experimental setup, cells have a finite thickness, therefore it is important to
investigate the time/length scales at which the semi-infinite solutions are applicable to
finite domain problems. Considering equation (2.58), at distance x L , it takes a time t
larger than 20.13 / pL D for the pore pressure to reach 5% of its maximum value at the
loaded boundary. Thus at times less than 20.13 / pL D the pore pressure perturbation at
the boundary 0x weakly influences the pore pressure at x L . In other words, at
distances beyond 3 pD t the pore pressure is less than 5% of its maximum at the
65
boundary. In conclusion, at short times, perturbations applied to the surface boundary are
confined to small regions near the boundary and thus the solution for the confined
problem can be estimated from semi-infinite solutions.
Figure 2-5 Creep response of a finite domain.
Top: Settlement response of the surface following a sudden application of aves = 250 Pa for a poroelastic
material with sE = 1 kPa, sn = 0 and pD = 40 µm2s-1. The dashed line is the semi-infinite response and the
lines are the solutions of equation (2.53) for poroelastic materials of thicknesses ranging from h = 2 to 8 µm.
Bottom: The displacement error considering the semi-infinite solution as the exact solution for the finite
thickness domain.
Furthermore, in relation to real AFM indentation experiments, to investigate the time
scales at which the half-space approximation can be employed, representative values of
the physical parameters derived from experimental data were used to compare the surface
66
displacements ( 0x ) in equations (2.53) and (2.56) for finite and semi-infinite
domains. As will be described later in Chapter 4, application of F ~ 6 nN via a spherical
tip with radius R = 7.5 µm results in indentation depths of 0d ~ 1 µm which leads to an
average stress of /ave F Rs p d ~ 250 Pa. Figure 2-5 shows the displacement as a
function of time considering thicknesses H = 2 - 8 µm for a poroelastic material. It can
be observed from this figure that for very small thicknesses (H < 3 µm) the semi-infinite
solution is applicable only at very short times ( t < 100 ms). For thicknesses of H ~ 5
µm (as shall be seen later this is approximately the height of the cells studied in this
thesis) the semi-infinite approximation is valid for the first ~ 300 ms of the process. The
half-space solution is a very good approximation for longer times ( t ~ 1 s) if the
thickness is H > 8 µm.
Semi-infinite domain stress relaxation Let us consider again the semi-infinite
problem but with sudden application of displacement (0, ) ( )u t UH t instead of load on
the top surface of a poroelastic material. This is the case of stress relaxation where the
pressure is suddenly increased due to fast compression of the top layer followed by
relaxation over time. Considering the far field conditions / 0u p x at x ,
the strain diffusion equation (2.32) transforms into a displacement diffusion equation
2
2.
x x
P
u uD
t x (2.59)
The displacement field satisfying the boundary conditions can be found as
( , ) erfc .2x
p
xu x t U
D t (2.60)
The corresponding pressure field becomes
2
4( , ) 1 p
xp D tDU
p x t eK tp
(2.61)
67
In Figure 2-6 the normalized pore pressure and the displacement field are plotted as a
function of depth at different times for 1pD .
Figure 2-6 Stress relaxation response of a poroelastic half-space.
Pore pressure (left) and displacement (right) solutions from the semi-infinite domain stress relaxation
problem. The curves are plotted as a function of depth at different times for 1pD .
2.5 Indentation of a poroelastic material
Conventional engineering methods such as compression, tensile and shear tests have
limitations for the characterisation of very soft materials due to difficulties in
manipulating soft and very small samples. Furthermore these methods do not have the
necessary accuracy and sensitivity to study the time-dependent dynamics of very soft
materials with short time scale responses. One of the best methods for characterisation of
very soft materials, especially cells, is an indentation test and as explained later in chapter
3, I conducted AFM dynamic micro-indentation experiments to study cell rheology in the
framework of poroelasticity. For indentation of a purely elastic material, there presently
exist several closed-form formulations such as Hertz’s equation (2.17) for obtaining
cellular elastic properties such as the shear modulus. Some closed-form expressions have
also been proposed for characterisation of time-dependent viscoelastic materials such as
indentation of a Zener viscoelastic material as presented in Chapter 3.7.3 . However due
to the higher complexity of the poroelastic equations, geometry and boundary conditions,
68
there exists no simple closed-form analytical solution that can be used to fit experimental
data. Therefore in this work an empirical expression derived from finite element (FE)
simulations [120] was employed to fit experimental poroelastic relaxation responses in
AFM indentation tests.
Figure 2-7 Spherical indentation and force relaxation a poroelastic material of finite thickness.
The indentor is pressed into the material until it reaches a target indentation depth of d . The force required to
induce such an indentation depth is iF applied with a short rise-time of rt . The applied force iF relaxes to
fF with the poroelastic characteristic time of t . (B) Compression of the solid matrix of a poroelastic
material with a short rise time rt t , pressurise the interstitial fluid. This pressurisation induces fluid
permeation expanding in a semi-spherical shape. The fluid migration can be affected by the substrate and for
h a the shape of expansion changes from a semi-spheroid to a 2D tube. (C) Experimental force-
relaxation data (grey dots) fitted with empirical solutions for force-relaxation of poroelastic materials
assuming the cell has a finite height (black) or that the cell can be approximated as a semi-infinite half-plane
(blue). Both empirical solutions fitted the experimental data well ( 2r > 0.95).
69
2.5.1 Indentation of a thin layer of poroelastic material
Very recently, detailed finite-element simulations of spherical indentation of thin layers
of poroelastic material (Figure 2-7) have been reported [121]. Recalling the Hertz
equation (2.17) for indentation of a semi-infinite elastic material, the reactive force on the
indentor can be expressed in terms of shear modulus G , Poisson ratio n and indentation
depth d : 1/2 3/2(8/3) /(1 ).HertzF GR d n For indentation of a poroelastic material of
finite thickness, the stress field is influenced by the substrate and thus the above equation
can be corrected for the effects of the substrate:
1/2 3/28
/ ,3 1 p
GF R f R hd d
n (2.62)
where pf is solved numerically and takes the following empirical form [121]:
20.46 0.82 / 2.36 /
/0.46 /p
R h R hf R h
R h
d dd
d (2.63)
For infinitely fast indentations ( rt t ) the fluid does not have enough time to
permeate through deformed regions and the poroelastic material behaves as an
incompressible elastic material (undrained condition) with the force response of
1/2 3/216
/ .3i pF GR f R hd d At long time scales, when the interstitial fluid has fully
redistributed (fully drained condition), the force imposed by the indenter is balanced by
the stress in the elastic porous matrix only. Under this condition, the force applied for the
prescribed indentation is 1/2 3/28
/3 1f p
s
GF R f R hd d
n. Thus, the Poisson ratio of
the solid matrix sn determines the ratio of forces at short and long timescales
/ 2 1i f sF F n . In intermediate regimes, the fluid permeates through pores and
force-relaxation can be approximated in terms of stretched exponential functions:
( )
expf
i f
F t F
F Fbat (2.64)
70
where 2
pD t
at is the poroelastic characteristic time required for force to relax from iF
to fF , and a and b are functions estimated empirically from the simulations that
depend on the contact radius a Rd and the height h of the layer:
2 3
4
2 3
4 5
1.15 0.44( / ) 0.89( / ) 0.42( / )
0.06( / )
0.56 0.25( / ) 0.28( / ) 0.31( / )
0.1( / ) 0.01( / ) .
R h R h R h
R h
R h R h R h
R h R h
a d d d
d
b d d d
d d
(2.65)
In my experiments, a typical height of h ~ 4.5 µm (at indention point) was measured for
HeLa cells and a and b were estimated from the above equations. Fitting experimental
force curves with the proposed stretched exponential function indicated that for h > 4.5
µm and d < 1.4 µm, approximating the cell to an infinite half-plane resulted in a less than
25% overestimation of the poroelastic diffusion constant pD (Figure 2-7C).
71
Chapter 3
Materials and Methods
Equation Chapter (Next) Section 1
3.1 Cell culture, generation of cell lines, transduction, and molecular
biology
In my experiments, I used three cell lines: cervical cancer HeLa cells, Madin-Darby
Canine Kidney epithelial (MDCK) and HT1080 fibrosarcoma cells. HeLa cells, HT1080
and MDCK cells were cultured at 37°C in an atmosphere of 5% CO2 in air in DMEM
(Gibco Life Technologies, Paisley, UK) supplemented with 10% FCS (Gibco Life
Technologies) and 1% PS. Cells were plated onto 50 mm glass bottomed Petri dishes
(Fluorodish, World Precision Instruments, Milton Keynes, UK). For MDCK cells, they
were cultured until forming a confluent monolayer. Prior to the experiment, the medium
was replaced with Leibovitz L-15 without phenol red (Gibco Life Technologies)
supplemented with 10% FCS.
To enable imaging of the cell membrane, we created a stable cell lines expressing the PH
domain of Phospholipase Cδ tagged with GFP (PHPLCδ-GFP), a phosphatidyl-inositol-
72
4,5-bisphophate binding protein that localises to the cell membrane. Briefly, PH-PLCδ-
GFP (a kind gift from Dr Tamas Balla, NIH) was excised from EGFP-N1 (Takara-
Clontech Europe, St Germain en Laye, France), inserted into the retroviral vector
pLNCX2 (Takara-Clontech), and transfected into 293-GPG cells for packaging (a kind
gift from Prof Daniel Ory, Washington University [122]). Retroviral supernatants were
then used to infect wild type HeLa cells, cells were selected in the presence of 1 mg.ml-1
G418 (Merck Biosciences UK, Nottingham, UK) for 2 weeks, and subcloned to obtain a
monoclonal cell line. Using similar methods, we created cell lines stably expressing
cytoplasmic GFP for cell volume estimation, GFP-actin or Life-act Ruby ([123]; a kind
gift of Dr Roland Wedlich-Soldner, MPI-Martinsried, Germany) for examination of the
F-actin cytoskeleton, GFP-tubulin for examination of the microtubule cytoskeleton, and
GFP-Keratin 18 (a kind gift of Dr Rudolf Leube, University of Aachen, Germany) for
visualisation of the intermediate filament network. HT1080 cells expressing mCherry-
LifeAct and MDCK cells expressing PHPLCδ-GFP were generated using similar
methods. The EGFP-10x plasmid was described in [124] and obtained through Euroscarf
(Frankfurt, Germany). Cells were transfected with cDNA using lipofectamine 2000
according to the manufacturer’s instructions the night before experimentation*.
3.2 Fluorescence and confocal imaging
In some experiments, we acquired fluorescence images of the cells being examined by
AFM using an IX-71 microscope interfaced to the AFM head equipped with an EMCCD
camera (Orca-ER, Hamamatsu, Germany) and piloted using µManager (Micromanager,
Palo-Alto, CA). Fluorophores were excited with epifluorescence and the appropriate filter
sets and images were acquired with a 40x dry objective (NA = 0.7). For staining of the
nucleus, cells were incubated with Hoechst 34332 (1 µg/ml for 5 min, Merck-
Biosciences).
*The cell lines described in this paragraph were generated with kind help from Dr. Charras.
73
In some cases, an AFM interfaced with a confocal laser scanning microscope (FV1000,
Olympus) was utilized to image the cellular indentation in the zx-plane. Images were
acquired with a 100x oil immersion objective lens (NA = 1.4, Olympus). Latex beads
attached to AFM cantilevers were imaged by exciting with a 647 nm laser and collecting
light at 680 nm. GFP tagged proteins were excited with a 488 nm laser and light was
collected at 525 nm. zx-confocal images passing through the centre of the bead were
acquired with 0.2 µm steps in z to give a side view of the cell before and after indentation
(Figure 3-2D).
3.3 Cell volume measurements
To measure changes in HeLa cell volume in response to osmotic shock, confocal stacks
of cells expressing cytoplasmic GFP were acquired at 2 min intervals using a spinning
disk confocal microscope (Yokogawa CSU-22, Yokogawa, Japan) with 100x oil
immersion objective lens (NA = 1.4, Olympus) and a piezo-electric z-drive (NanoscanZ,
Prior, Scientific, Rockland, MA). Stacks consisted of 40 images separated by 0.2 µm and
were acquired every 2 min for a total of 30 mins. Exposure time and laser intensity were
set to minimize photobleaching. A custom written code in Matlab (Mathworks Inc,
Cambridge, UK) was used to process the z-stacks and measure the cell volume at each
time step. Briefly, the background noise of stack images was removed, images were
smoothed, and binarised using Matlab Image Processing Toolbox functions. Following
binarisation, series of erosion and dilatation operations were performed to create a
contiguous cell volume image devoid of isolated pixels (Figure 3-1A, B). The sum of
nonzero pixels in each stack was multiplied by the volume of a voxel to give a measure of
cell volume at each time step. All experiments followed the same protocol: five stacks
were captured prior to change in osmolarity and then cell volume was followed for a
further 25 minutes (see Figure 5-2A).
74
Figure 3-1 Cell volume measurement.
(A, B) The volume of HeLa cells was measured by acquiring fast xyz-t confocal stacks. (A) Image of HeLa
cytoplasmic GFP cell showing the bottom of the cell before (I) and after (II) application of hyperosmotic
shock induced by PEG-400. The decrease in cell volume increases the concentration of GFP molecules
resulting in an increase in fluorescence intensity. (B) Reconstruction of cell profile from confocal z-stacks
before I and after II application of PEG-400. (C-I) xy image of MDCK monolayer expressing PH-PLCδ. (C-
II) zx section of MDCK cells expressing a membrane marker (PH-PLCδ, green) and stained for nucleic acids
with Hoechst 34332 (blue). Nuclei localised to the basal side of cells far from the apex. (C-III) zx profile of
MDCK cells before (green) and after (red) application of 500 mM sucrose. Cell height decreased significantly
in response to increase in medium osmolarity. Scale bars = 10µm.
The volume of PH-PLCδ MDCK cells in response to osmotic perturbations was estimated
by measuring the heights of cell from confocal z-slice images after and before application
osmotic shocks (Figure 3-1C). During the osmotic perturbations, I verified that the xy
projected area of the cells and the xy positions of cell-cell junctions were not perturbed
significantly and therefore measuring the change in cell height was a good approximation
for estimating changes in cell volume.
Changes in extracellular osmolarity were effected by adding a small volume of
concentrated sucrose, 400-Dalton polyethylene glycol (PEG-400, Sigma-Aldrich, [125]),
75
or water to the imaging medium. When increasing osmolarity by addition of osmolyte,
MDCK cells were treated with EIPA (50 µM, Sigma-Aldrich), an inhibitor of regulatory
volume increases [126]. When decreasing the osmolarity by addition of water, the cells
were simultaneously treated with the inhibitors of regulatory volume decreases, NPPB
(200 μM; Tocris, Bristol, UK) and DCPIB (50 µM, Tocris), to achieve a sustained
volume increase [126]. Cells were incubated with these inhibitors for 30 minutes prior to
the addition of osmolytes (PEG-400, sucrose or water).
3.4 Disrupting the cytoskeleton
3.4.1 Pharmacological treatments
Cells were incubated in culture medium with the relevant concentration of drug for 30
min prior to measurement. The medium was then replaced with L-15 with 10% FCS plus
the same drug concentration such that the inhibitor was present at all times during
measurements. Cells were treated with latrunculin B (to depolymerise F-actin, Merck-
Biosciences), nocodazole (to depolymerise microtubules, Merck-Biosciences), and
paclitaxel (to stabilize microtubules, Merck-Biosciences), and blebbistatin (to inhibit
myosin II ATPase, Merck-Biosciences).
3.4.2 Genetic treatments
To examine the effect of uncontrolled polymerization of cytoplasmic F-actin, Dr.
Moulding transduced HeLa cells stably expressing Life-act ruby with lentivirus encoding
WASp I294T as described in [26]. Lentiviral vectors expressing enhanced GFP fused to
human WASp with the I294T mutation were prepared in the pHR’SIN-cPPT-CE and
pHR’SIN-cPPT-SE lentiviral backbones as described previously [26, 127]. Lentivirus
was added to cells at multiplicity of infection of 10 to achieve approximately 90%
transduction.
76
To examine the perturbations to cytoskeletal proteins, Dr. Charras generated the plasmids
described below, and I purified the cDNA and transfected the cells. To examine the effect
of uncontrolled polymerisation of tubulin, I transfected HeLa cells with a plasmid
encoding γ-tubulin-mCherry, a microtubule nucleator [128, 129]. To disrupt the keratin
network of HeLa cells, keratin 14 R125C-YFP (a kind gift from Prof Thomas Magin,
University of Leipzig) was overexpressed, a construct that acts as a dominant mutant and
results in aggregation of endogenous keratins [130]. To disrupt F-actin crosslinking by
endogenous α-actinin, cells were transfected with a deletion mutant of α-actinin lacking
an actin-binding domain (ΔABD-α-actinin, a kind gift of Dr Murata-Hori, Temasek Life
Sciences laboratory, Singapore). Cells were transfected with cDNA using lipofectamine
2000 the night before experimentation.
3.5 Visualising cytoplasmic F-actin
To visualise cytoplasmic F-actin density, cells were fixed for 15 minutes with 4% PFA at
room temperature, permeabilised with 0.1% Triton-X on ice for 5 min, and passivated by
incubation with phosphate buffered saline (PBS) and 10 mg/ml bovine serum albumin
(BSA) for 10 min. They were then stained with Rhodamine-Phalloidin (Invitrogen) for 30
min at room temperature, washed several times with PBS-BSA, and mounted for
microscopy examination on a confocal microscope.
77
Figure 3-2 AFM experimental setup.
(A) Schematic diagram of the experiment. (A-I) The AFM cantilever is lowered towards the cell surface with
a high approach velocity approachV ~ 30 µm.s-1. (A-II) Upon contacting the cell surface, the cantilever bends
and the bead starts indenting the cytoplasm. Once the target force MF is reached, the movement of the
piezoelectric ceramic is stopped at MZ . The bending of the cantilever reaches its maximum. This rapid force
application causes a sudden increase in the local stress and pressure. (A-III) and (A-IV) Over time the cytosol
in the indented area redistributes inside the cell and the pore pressure dissipates. Strain resulting from the
local application of force propagates through the elastic meshwork and at equilibrium; the applied force is
entirely balanced by cellular elasticity. Indentation (I-II) allows the estimation of elastic properties and
relaxation (III-IV) allows for estimation of the time-dependent mechanical properties. In all panels, the red
line shows the light path of the laser reflected on the cantilever, red arrows show the change in direction of
the laser beam, black arrows show the direction of bending of the cantilever, and the small dots represent the
propagation of strain within the cell. (B) Temporal evolution of the cantilever position and force applied
during the experiment. Before the cantilever contacts the cell, no force is applied (I). After contact, force
increases until the target force (II). Finally, force relaxes (III) until equilibrium (IV). (C) Schematic drawing
showing the zone of indentation. A bead of radius R is pressed into the cell surface creating a depression of
radius a and depth ad . The compression of the cell surface by the bead induces a pressure increase that
squeezes a volume 2~ ad of fluid out of the indentation region (arrows). (D) Z-x confocal image of a HeLa
cell expressing PH-PLCδ1-GFP (a membrane marker) corresponding to phases (I) and (IV) of the experiment
described in A. The fluorescent bead attached to the cantilever is shown in blue and the cell membrane is
shown in green. Scale bar = 10 μm.
78
3.6 Atomic force microscopy indentation experiments
Force-distance and force-relaxation measurements were acquired with a JPK
Nanowizard-I (JPK instruments, Berlin, Germany) interfaced to an inverted optical
fluorescence microscope (IX-81 or IX-71, Olympus). For most experiments, AFM
cantilevers (MLCT, Bruker, Karlsruhe, Germany) were modified by gluing beads to the
cantilever underside with UV curing glue (UV curing, Loctite, UK). Cantilever spring
constants were determined prior to gluing the beads using the thermal noise method
implemented in the AFM software (JPK SPM, JPK instruments). Prior to any cellular
indentation tests, the sensitivity of the cantilever was set by measuring the slope of force-
distance curves acquired on glass regions of the petri dish. For measurements on cells, I
used cantilevers with nominal spring constants of 0.01 N.m-1
and fluorescent latex beads
with radii of 7.5 μm (ex645/em680, Invitrogen). For measurements on hydrogels, I used
cantilevers with nominal spring constants of 0.6 N.m-1
and glass beads with radii of 25
μm (Sigma).
During AFM experiments, the bead on the cantilever was aligned over regions near the
cell nucleus and measurements were acquired in several locations in the cytoplasm
avoiding the nucleus (except when doing spatial mapping, see section 3.7.5). To
maximize the amplitude of stress relaxation, the cantilever tip was brought into contact
with the cells using a fast approach speed ( approachV > 10 µm.s-1
) until reaching a target
force MF (Figure 3-2A-I, A-II, B). With these settings, force was applied onto the cells in
less than 35 ms on average, which is fast compared to the expected poroelastic relaxation
time (~0.1-10 s, see Chapter 4 section 1.1). Hence, in the analysis of stress relaxation I
assumed that force application was quasi-instantaneous. Upon reaching the target force
MF the piezoelectric ceramic length was kept constant at MZ and the force-relaxation
curves were acquired at constant MZ (Figure 3-2A-III, A-IV). After 10 s, the AFM tip
was retracted with the same speed as the approach.
79
During force-relaxation measurements, the z-closed loop feedback implemented on the
JPK Nanowizard was used to maintain a constant z-piezo height. When acquiring force-
relaxation curves with high approach velocities ( approachV > 10 µm.s-1
), the first 5 ms of
force-relaxation were not considered in the data analysis due to the presence of small
oscillations in z-piezoelectric ceramic height immediately after contact due to the PI
feedback loop implemented in the JPK software. When acquiring force-relaxations curves
for averaging over multiple cells, I used approach velocities approachV ~ 10 µm.s-1
that did
not give rise to oscillations in length of the z-piezoelectric ceramic.
3.6.1 Measuring indentation depth and cellular elastic modulus
The approach phase of the AFM force-distance curves (Figure 3-3) was analysed to
extract the bulk elastic modulus E , which is linearly related to the shear modulus G
through the Poisson ratio via equation (2.4); 2 (1 )E G n . During indentation (after
contact) the force the cantilever is related to the cell stiffness and the deflection of the
cantilever d by
0( ),c cF k d k d d (3.1)
where ck is the cantilever’s spring constant, d is the deflection of cantilever (correlated
to changes in reflection of AFM laser beam which is detected by photo-detector), and 0d
is the deflection offset at the point of contact where the force is zero.
The indentation depth d is calculated by subtracting the cantilever deflection d from
the piezo translation z
0 0
0 0 0
( ) ( )( ) ( ) ,
z d z z d dz d z d w w
d (3.2)
where 0z is the translation of the piezo at the contact point, ( )w z d and
0 0 0w z d are the transformed variables. E was then estimated by fitting the force-
indentation data with a Hertzian contact model between a sphere and an infinite half
80
space (equation (2.17)). In this framework, the relationship between the applied force F
and the indentation depth d is:
1/2 3/22
4.
3 1E
F R dn
(3.3)
with n the Poisson ratio, and R the radius of the spherical indenter. The above Hertz
model of indentation can be used to compute the elastic modulus of a linear elastic
material of infinite thickness (see Chapter 2.1.1). To limit the effects of finite cell
thickness, in my analysis I only considered the force-indentation/relaxation curves in
which the indentation depths were less than 25% of the cell height. This ensures errors of
less than 20% in the estimation of the elastic modulus (see equation 2.63) in my
experiments. To use the Hertz model in this work I assumed the cell to be a isotropic,
homogeneous and linear elastic materials and the contact surfaces were assumed to be
frictionless and non-adhesive [131]. Although these assumptions are not fully accurate for
indentation of adherent cells, the Hertz model is still widely used for characterisation of
cells and soft materials providing fairly accurate measurements of elastic modulus [132].
Figure 3-3 AFM force-distance curve.
The approach phase of AFM force-distance curve (Figure 3-2A-I and II). The force distance AFM data
(dotted points) is fitted with two linear (blue line) and nonlinear (red line) curves for the noncontact and
contact portions of the curve respectively. The contact point is estimated by choosing the point that minimises
the total mean square error of the fitted curves.
81
Together with equations (3.1) and (3.2), contact mechanics models (like equation (3.3))
can be fit to AFM indentation data. However the main challenge in analysing AFM force-
distance curves is to design an automated routine to find the correct contact point which
enables us to compute the indentation depth. The contact point between the cell and the
AFM tip was estimated using the method described in [133] implemented in our
laboratory, see Figure 3-3. In brief, as a first step the force distance-curve is transformed
into a d -w curve introducing the variable ( )w z d . Then our custom written Matlab
code marches along all points of the d -w curve, and assumes each point is the potential
contact point. Considering ( 0iw , 0
id ) as the contact point, at each iteration i the initial
portion of the curve (assumed to be the noncontact region) was fitted with a line while the
latter portion (assumed to be the portion of the curve in contact) was fitted with a power
function as in equation (3.3):
3/20 0( ) ( ) ,i i id d b w w (3.4)
where ib is the sole fitting parameter. The total mean square error (MSE) from both fits
was calculated and the contact point ( 0nw , 0
nd ) was chosen as the point where the total
MSE reaches its minimum value at i n with 1/2 24 / 3(1 )ncb ER kn (see
Figure 3-3 ).
Comparing the long time-scale and short time-scale limits allows for estimation of the
Poisson ratio of the solid matrix; (0)/ ( ) 2(1 )sF F n . However for cellular
indentations the force relaxation curves do not reach a relaxation plateau for estimation of
( )F and thus for estimation of the elastic modulus, I assumed a Poisson ratio of sn
=0.3.
3.6.2 Measurement of the poroelastic diffusion coefficient
A brief description of the governing equations of linear isotropic poroelasticity and the
relationship between the poroelastic diffusion constant pD , the shear modulus G , and
82
hydraulic permeability K are given in Chapter 2 section 3. To experimentally measure
the cellular poroelastic properties, I monitored stress-relaxation following rapid local
application of force by AFM microindentation (Figure 3-3, Figure 3-4 and Figure 5-1A). I
analysed my experiments as force-relaxation in response to a step displacement of the cell
surface. No closed form analytical solution for indentation of a poroelastic infinite half
space by a spherical indentor exists. However, an approximate solution obtained by FE
simulations gives [120]:
0.908 1.679( )
0.491e 0.509e ,f
i f
F t F
F Ft t (3.5)
where /( )pD t Rt d is the characteristic poroelastic time required for force to relax
from iF to fF . Cells have a limited thickness h and therefore the infinite half-plane
approximation is only valid at time-scales shorter than the time needed for fluid diffusion
through the cell thickness: 2~ /hp pt h D . As we shall see in the next chapters, in
experiments on HeLa cells, I measured h ~ 5 µm and pD ~ 40 µm2.s
-1 setting a time-
scale hpt ~ 0.6 s. I confirmed numerically that for times shorter than ~ 0.5 s,
approximating the cell to a half-plane gave errors (calculated by comparing the half space
approximation and the finite thickness solution) of less than 20% (Figure 2-5 and
Figure 2-7C). For short time-scales, both terms in equation (3.5) are comparable and
hence as a first approximation, the relaxation scales as ~e t . Equation (3.5) was utilized
to fit our experimental relaxation data, with pD as single fitting parameter and I fitted
only the first 0.5 s of relaxation curves to consider only the maximal amplitude of
poroelastic relaxation and minimise errors arising from finite cell thickness.
3.6.3 Determining the apparent cellular viscoelasticity
To determine the single phase viscoelastic parameters, I modelled cells using a standard
linear model consisting of a spring-damper (stiffness 1k and apparent viscosity h ) in
parallel with another spring (stiffness 2k ), as described in [53] (see Figure 2-2). In this
83
model, the applied force decays exponentially when the material is subjected to a step
displacement at t = 0:
1( ).
k tf
i f
F t Fe
F Fh (3.6)
The spring constant 1k scales with the elastic modulus 1 ~k E (estimated from approach
phase of indentation tests) and therefore the experimental force relaxation curves were
fitted using equation (3.6), with h as the sole fitting parameter.
3.6.4 Measuring cell height
To measure cell height at each contact point, I collected a force-distance curve on the
glass substrate next to the cell being examined. The 0z calculated from the force-distance
curve on the glass was used as a height reference. As shown in Figure 3-4, at each
indentation point, the cell height was estimated by subtracting the 0z of the cell from the
reference 0z .
Figure 3-4 Calculating the height of cell at contact point.
A force-distance curve acquired on the glass substrate next to the cell examined was used as the reference to
estimate the height of the cell at the contact point
84
3.6.5 Spatial probing of cells
To investigate spatial variations in the cellular poroelastic properties, I acquired
measurements at several locations on the cell surface along the long axis of the cell with a
target force of MF = 4 nN. To locate indentation points in optical images of the cell, a
fluorescence image of the bead resting on the glass surface close to the cell was acquired,
and the centre of the bead was used as a reference point. This reference point was used to
estimate the height of the cell for each measurement, as well as the position of indentation
on the cell surface, the xy-coordinates of each stress relaxation measurement being
known from the AFM software. The position of the measurement point relative to the
nucleus was estimated by acquiring a fluorescence image of the nucleus and calculating
the centre of mass of the nucleus (see Figure 3-5). This also enabled us to know whether
the measurement point was above the nucleus or not.
85
Figure 3-5 Spatial AFM measurement
(I) Representative phase contrast image of a HeLa cell and the location of AFM measurements on its surface.
The position of the height reference, where the AFM tip contacts the glass surface was also chosen as the xy
reference (fluorescence image of the bead, IV). The location of the nucleus was determined using Hoechst
34332 staining and the centroid of the nucleus (red dot in I) was used as a reference for displaying the
measured properties. To determine the exact pathline of the bead and to calculate the distance of each point
from the centre of nucleus, images of the cell (II), the cell nucleus (III), and the bead (IV) were overlaid (I).
Scale bars = 10 μm.
3.7 Microinjection and imaging of quantum dots
PEG-passivated quantum dots (qdots 705, Invitrogen) were diluted in injection buffer (50
mM potassium glutamate, 0.5 mM MgCl2, pH 7.0) to achieve a final concentration of 0.2
μM and microinjected into HeLa cells as described in [134]. Quantum dots were imaged
on a spinning disk confocal microscope by exciting at 488 nm and collecting emission
above 680 nm. To qualitatively visualize the extent of quantum dot movement, time
series were projected onto one plane using ImageJ.
86
3.8 Rescaling of force relaxation curves
Plotting the obtained experimental relaxation curves in different scale coordinates and/or
normalising them in different ways can provide useful insights into the functional nature
of these curves. For instance it is difficult to differentiate between exponential relaxation
curves and weak power law curves unless the curves are plotted in log-log coordinates.
Furthermore, to differentiate between poroelastic and viscoelastic responses during
indentation tests several methods for normalisation of relaxation curves have been
proposed [120, 135-137]. One successfully applied method is based on the collapse of
normalised force relaxation curves acquired with different indentation depths onto a
single master curve [137]. Prompted by this, I have proposed the following more general
force normalisation method:
( ) ( *)
,( 0) ( *)F t F t tF t F t t
(3.7)
where *t can be any arbitrary time. Let us consider that the force relaxation is decaying
function of the form 1 2( ) ( , , ,..., )nF t F t t t t that is a function of a number of (finite or
infinite) characteristic relaxation times it . For force relaxation curves acquired with
different indentation depths, if the characteristic relaxation times (or in general the decay
function F ) do not have any form of functional dependency on indentation depth (as
during linear viscoelastic or power law responses), force normalisation using equation
(3.7) results in the collapse of all the relaxation curves onto a single curve. In contrast, the
characteristic relaxation time in poroelastic materials depends on the indentation depth
and force-normalisations alone will not result in collapse of curves onto a single master
curve.
Considering the spherical indentation of a purely poroelastic material, the force relaxation
can be written in the form ( ) [ (0) ( )] ( / ) ( )pF t F F P D t R Fd where P is the
poroelastic response function (decaying in an exponential manner from 1 at 0t to 0 at
t ) and (0)F and ( )F are the forces at very short (t = 0+) and very long (t = ∞)
87
timescales, respectively. For any given indentation depth d , using the normalisation in
equation (3.7) results in:
( / ) ( * / )( ) ( *)
( 0) ( *) 1 ( * / )p p
p
P D t R P D t RF t F t tF t F t t P D t R
d dd
(3.8)
To obtain collapse of force relaxation curves onto a single master curve for any *t using
the above equation time must be rescaled with indentation depth. First, *t must be
selected so that it is proportional to the indentation depth d specific to each curve:
*t ad where a (with unit of s. μm-1
) is an arbitrary number fixed for all curves. Then
rescaling of time with respect to indentation depth ( /t d ), leads to collapse of all
experimental curves onto one master curve.
Note that, if the chosen *t is such that * / pt R Dt d , then the force relaxation
curve reaches a plateau: ( ) ( )F t F and ( * / ) 0pP D t Rd and equation (3.7)
becomes:
( ) ( *)
( / ),( 0) ( *) p
F t F t tP D t R
F t F t td (3.9)
and therefore rescaling of time with respect to indentation depth ( /t d ), is sufficient to
obtain collapse of all relaxation curves as in [137].
3.9 Polyacrylamide hydrogels
For comparison with cells, I acquired force-relaxation measurements on polyacrylamide
(PAAm) hydrogels, which are well-characterised poroelastic materials. For the
measurements, we made gels of 15% acrylamide-bis-acrylamide crosslinked with
TEMED and ammonium persulfate following the manufacturer’s instructions (Bio-Rad,
UK).
88
3.10 Particle tracking using defocused images of fluorescent beads
Optical images of fluorescent beads acquired by wide-field microscopy can exhibit
different shapes depending on the distance between the bead and the plane of focus. The
in-focus image of a bead is a spot with a certain diameter, while when the plane of focus
is away from the bead centre by a distance of a few micrometres, sets of concentric rings
are visualised (see Figure 3-6A). The z-distance of the particle from the imaging plane is
precisely correlated to the radius of the outer ring. As a very simple 3D particle tracking
technique, calculating the centre and radius of the outer ring over several frames over
time can provide a very precise means for tracking the coordinates of a bead: ( , , , )x y z t
[138]. This technique is presented in Chapter 4.2 and was employed to study the
swelling/shrinking kinetics of cell (Figure 4-7).
3.10.1 Experimental protocol and calibration setup
Yellow-Green carboxylate-modified fluorescent microspheres 0.5 µm in diameter
(FluoSpheres, Molecular Probes, Invitrogen) were coated with collagen-I following the
protocol suggested by the manufacturer. To attach beads to the cell membrane prior to
experiments, Life-act ruby HeLa cells were incubated for ~ 30 min with a dilute solution
(100x dilution) containing fluorescent collagen coated beads.
A Piezo-electric z-stage (NanoscanZ, Prior, Scientific, Rockland, MA) was used to
calibrate defocused images and to estimate the height of the cell at the location of the
attached beads (Figure 3-6). The z-stage was fixed on top of the optical fluorescence
microscope stage (IX-71, Olympus) and was piloted using µManager (Micromanager,
Palo-Alto, CA) to step up/down while taking epifluorescence images. To find the
relationship between the z-positions of a bead and the radius of the outer rings formed in
the defocused images, fluorescence z-stack images of the bead with a step size of 100 nm
were acquired. To measure the poroelastic diffusion constant using equation (4.6), the
height of the bead also needed to be determined. For this, the z-position of the bead with
89
respect to the bottom of the cell was estimated by acquiring fluorescence z-stack images
of Lifeact-Ruby HeLa cells with step size of 200 nm over large distances (~ 10 μm,
starting from a few micrometres above the cell and ending a few micrometres below the
cell-glass interface). The height of the cell was measured as the distance between the
plane that shows an in-focus image of the stress fibres at the bottom of the cell and the
plane focused on the bead (Figure 3-6D).
Figure 3-6 High resolution defocusing microscopy of fluorescent bead.
(A) Images of a 500 nm fluorescent bead attached to the cell membrane taken at different distances from bead
centre appears as concentric rings. The radius of the outermost ring in the image can be related to the distance
between the bead centre and the plane of imaging. (B) The averaged radial intensity curves calculated when
the imaging plane is moved in 100 nm increments away from the bead focal plane. The radius of outer rings
can be estimated using these curves. Inset is the normalised averaged radial intensity curves. To determine the
radius of outer rings, the peak of each curve can be fitted with a Gaussian function. (C) The radial position of
the outer ring is linearly related to the distance between the imaging plane and the bead focal plane. Using
this concept the z-position of a particle can be tracked with very high accuracy (~ 10 nm). (D) The height of a
bead attached to the cell is estimated by focusing on the bottom of the dish in (I) where stress fibres are in
focus, and on the bead attached to the cell membrane in (II).
90
3.10.2 Radial projection
The 2D defocused images of a bead are circularly symmetric so a 1D radius function was
used to detect the radius of the outer ring in each defocused image. Since there were
several beads attached to the cell membrane, the rough central position of the bead of
interest was selected visually and then the precise coordinates of the centroid of the
chosen bead ( , )c cx y were estimated by fitting a 2-D Gaussian function to the image. An
arbitrary annulus interval was set to generate a radial distance vector
, 0...ir i i N . By knowing the precise centre of the image, the distance of each
pixel in the image from the centroid was estimated. According to the radial distance of
each pixel d , there are total of iP pixels lying between radii 1ir and 1ir . For each pixel
within the annulus between radii 1ir and 1ir the averaged radial intensity vector ( )R ir
(see Figure 3-6B) was calculated by splitting the intensity of each pixel I in this annulus
linearly according to its radial position d :
1
1
11
11
1( )
.
i
i
P
R i n nPn
nn
ii i
ni
i i
r w I
w
d rif r d r
w r dif r d r
(3.10)
To find an estimate with sub-pixel resolution for the radial position of the outer rings, two
approaches were implemented. In the first approach the radius of the outer ring was
calculated by finding the position of the outer peak in the averaged radial intensity curve
(using an inbuilt Matlab function) and fitting a rotationally symmetric Gaussian function
to the intensity curve neighbouring this point. In the second approach the averaged radial
intensity curve was normalised to its maximum intensity as shown in the inset of
Figure 3-6B and the intersection of the curve with the line y = 0.05 provided an estimate
for the radial position of the ring where the intensity of the outer ring diminishes to zero.
91
3.11 FRAP experiments
I performed the fluorescence recovery after photobleaching (FRAP) experiments
described in the following with kind help from Marco Fritzsche. FRAP experiments were
performed using a 100x oil immersion objective lens (NA = 1.4, Olympus) on a scanning
laser confocal microscope (Olympus Fluoview FV1000; Olympus). Fluorophores
including GFP-tagged proteins and CMFDA (a small cytoplasmic fluorescent probe, 5-
chloromethylfluorescein diacetate, Celltracker Green, Invitrogen) were excited at a
wavelength of 488 nm. For experiments with CMFDA, cells were incubated with 2 µM
CMFDA for 45 min before replacing the medium with imaging medium. For experiments
with cytoplasmic GFP or EGFP-10x, cells were transfected with the plasmid of interest
the day before experimentation. To obtain a strong fluorescence signal and minimize
photobleaching, a circular region of interest (ROI, 1.4 µm in diameter) in the middle of
the cytoplasm was imaged, setting the laser power to 5% and 1% of the maximum output
(488nm wave length, nominal output of 20mW) for GFP and CMFDA, respectively. Each
FRAP experiment started with five image scans to normalise intensity, followed by a 1s
bleach pulse produced by scanning the 488nm laser beam line by line (at 100% power for
GFP and 25% power for CMFDA) over a circular bleach region of nominal radius nr =
0.5 μm centred in the middle of the imaging ROI. To sample the recovery sufficiently
fast, an imaging ROI of radius of 0.7 μm was chosen allowing acquisition of fluorescence
recovery with a frame rate of 50 ms per frame.
92
Figure 3-7 Estimation of the effective bleach radius in HeLa cells FRAP experiments.
(I) Confocal image of cells loaded with the cytoplasmic indicator CMFDA (green). Nuclei (blue) were
stained with Hoechst 34332. Scale bar = 10 µm. (II-III-IV) Time series of a photobleaching experiment in the
cytoplasm of the cells shown in (I, indicated by the white circle). The white circle in images (II-IV) denotes
the nominal photobleaching zone. In all three images, scale bars = 1 µm. (II) shows the last frame prior to
photobleaching. (III) shows the first frame after photobleaching. The bleached region is clearly apparent and
larger than the nominal photobleaching zone. (IV) After a long time interval, fluorescence intensity re-
homogenises due to diffusion. (V) Fluorescence intensity immediately after photobleaching in the cytoplasm
centred on the photobleaching zone in frame (III). Fluorescence was normalised to the intensity far away
from the photobleaching zone.
93
3.11.1 Effective bleach radius
Most experiments were carried out on cytoplasmic fluorophores. Because of their rapid
diffusion within the cytoplasm during the bleaching process, the effective radius over
which photobleaching takes place ( er ) is larger than the nominal radius ( nr ). In my
experiments to measure the translational diffusion coefficient TD precisely, I needed to
first determine er . To experimentally estimate the effective radius er , I performed a
separate series of FRAP experiments on cells loaded with CMFDA in hyperosmotic
conditions. An imaging area larger than the effective photobleaching radius (4 µm in
diameter) was empirically chosen and photobleaching recovery in this area was imaged
with a frame rate of 0.2 s/frame. Next, using a custom written Matlab program, I
calculated the radial projection profile of the fluorescence intensity in the imaging region
for the first postbleach image ( , 1)I r t and then fitted the experimental intensity profile
with the bleach equation for a Gaussian laser beam to determine er (Figure 3-7):
2
2( ) exp exp 2 ,
e
rI r K
r (3.11)
where r is the radial distance from the center of bleach spot and K is the bleaching
constant related to the intensity of the bleaching laser and the properties of the
fluorophore.
3.11.2 FRAP analysis
FRAP recovery curves ( )I t were obtained by normalising the mean fluorescence intensity
within the nominal bleach area for each frame to the average fluorescence intensity of the
first prebleach image. The translational diffusion coefficient TD was estimated using the
model for a uniform-disk laser profile and ideal bleach [139, 140] following the methods
described in [141]:
2 2 2
0 1( ) exp ,8 8 8e e e
nT T T
r r rI t I I
D t D t D t (3.12)
94
where ( ) ( )/ ( )nI t I t I is the normalized fluorescence intensity in the bleached spot
as a function of time t , er is the effective bleach radius, 0I and 1I are modified Bessel
functions. In the experimentally acquired FRAP curves, the first post-bleach value
( 1 )nI t s was larger than zero (Figure 5-4 and Figure 5-5B, C), something that could be
due to rapid fluorophore diffusion during the bleach process or incomplete weak
bleaching [141]. In cells exposed to hyperosmotic conditions, the intensity of the first
post-bleach time point was over 30% lower than in isoosmotic conditions, suggesting that
the non-zero value of this first time point was due to rapid diffusive recovery. Based on
this observation, we assigned a timing of 0t 0.1 s to the first post bleach time point
when fitting fluorescence recovery curves. The validity of the fitting was verified by
comparing the translational diffusion constant of GFP and CMFDA estimated with our
technique to previously published values.
3.11.3 Estimation of cortical F-actin turnover half-time
To assess the turnover rate of the F-actin cytoskeleton, HeLa cells stably expressing
actin-GFP were blocked in prometaphase by overnight treatment with 100 nM
nocodazole. Under these conditions, cells form a well-defined actin cortex. To estimate F-
actin turnover, photobleaching was performed on the F-actin cortex and recovery was
imaged at a frame rate of 0.9s/frame (Figure 5-4B).
3.12 Data processing, curve fitting and statistical analysis
Indentation and stress relaxation curves collected by AFM were analysed using custom
written code in Matlab (Mathworks Inc, Cambridge, UK). Data points where the
indentation depth d was larger than 25% of cell height were excluded. Goodness of fit
was evaluated by calculating 2r values and for analyses only fits with 2r > 0.85 were
considered (representing more than 90% of the collected data). The calculated value for
95
each group of variables is presented in terms of mean and standard deviation (Mean±SD).
To test pairwise differences of population experiments, the student’s t-test was performed
between individual treatments. Values of p < 0.01 compared to control were considered
significant and are indicated by asterisks in the graphs.
96
Chapter 4
The cytoplasm of living cells
behaves as a poroelastic material
Equation Chapter (Next) Section 1
The cytoplasm of living cells is constituted of a porous elastic solid meshwork
(cytoskeleton, organelles, macromolecules, etc) bathing in an interstitial fluid (cytosol)
(see Figure 1-1 and Figure 1-10A). Based on this description, it is not very surprising that
the cytoplasm has many hydrogel-like characteristics. Hydrogels, an aggregate of cross-
linked polymer network and water molecules, are ubiquitous in nature and especially in
biological tissues. One of the main characteristics of hydrogels is that their mechanical
response is manifested through concurrent interaction of the solid network and the fluid
phase. In such fluid-filled sponges, the rate of relaxation is limited by the rate at which
interstitial fluid can redistribute within the solid network. As briefly described in Chapter
2.3, the Biot theory of poroelasticity has been applied to study the rheology of fluid-
infiltrated porous elastic solids. In this chapter, two different experimental approaches are
utilized to examine the poroelastic behaviours of cytoplasm and compare it to hydrogels.
97
First, by employing AFM micro indentation tests acquired on cells and hydrogels, the
functional form of force-relaxation curves at different timescales was examined. Then the
idea of poroelastic cytoplasm is validated by comparing the force relaxation curves
acquired on cells with those acquired on hydrogels as these are well-characterized
poroelastic materials. Second, to further test the poroelastic behaviour of cytoplasm, the
rheology of cells in response to different types of osmotic perturbation were studied and
the swelling/shrinking kinetics of cytoplasm were investigated in the framework of
poroelasticity.
4.1 laitanadfaateahfenfnennatedenadfafh feftIneadninatednI
4.1.1 Establishing the experimental conditions to probe poroelasticity
First, I established the experimental conditions under which water redistribution within
the cytoplasm (or hydrogel) might contribute to force-relaxation. In our experiments
(Figure 3-2), following rapid indentation with an AFM cantilever (3.5-6nN applied during
a rise time rt ~ 35ms resulting in typical cellular indentation of d ~ 1µm), force
decreased by ~ 35% whereas indentation depth only increased by less than ~ 5%,
therefore our experiments measure force-relaxation under approximately constant applied
strain (Figure 4-1A). Relaxation in poroelastic materials is due to water movement out of
the porous matrix in the compressed region (Figure 2-7B, Figure 3-2C). The time-scale
for water movement is 2~ /p pt L D (L is the length-scale associated with
indentation[120]: ~L Rd with R the radius of the indenter) and therefore poroelastic
relaxation contributes significantly if the rate of force application is faster than the rate of
water efflux: r pt t . Previous experiments estimated pD ~ 1-100 µm2.s
-1 in cells [100,
101] yielding a characteristic poroelastic time of pt ~ 0.1-10 s, far longer than rt . Hence,
if intracellular water redistribution is important for cell rheology, force-relaxation curves
should display characteristic poroelastic signatures for times up to pt ~ 0.1-10 s.
98
Figure 4-1 Stress relaxation of HeLa cells in response to AFM microindentation
(A) Temporal evolution of the indentation depth (black) and the measured force (grey) in response to AFM
microindentation normalized to their values when the target force is reached. Inset: approach phase from
which the elasticity is calculated (grey curve). The total approach time lasts less than 50 ms. (B) cell height
distribution at the indentation point. (C) Indentation depth distribution. In B and C, N indicates the total
number of measurements and n indicates the number of cells.
For hydrogels similar to the ones that were used in our experiments, poroelastic diffusion
constants of pD ~ 100 µm2.s
-1 have been reported [137]. Considering this estimate,
applying indention depths of d ~ 2 µm with the same size indentors as for cells (radius of
7.5 µm) would give characteristic poroelastic times pt < 0.1 s, too fast to accurately
observe poroelastic effects, given a rise times of rt ~ 0.1 s. Therefore to be able to
reliably observe poroelastic effects, I used a larger indentor ( radius of R = 25 µm).
Using this larger indentor with d ~ 2 µm gave an expected characteristic poroelastic time
of pt > 0.5 s much larger than the experimental rise time rt .
99
4.1.2 Poroelasticity is the dominant mechanism of force-relaxation in hydrogels
To test our rescaling strategies on hydrogels, force-indentation and force relaxation
curves were acquired on hydrogels by setting target forces of 40, 80 and 140 nN as shown
in Figure 4-2. For each set of target forces, 20 replicate measurements were performed on
a 100 μm × 100 μm area of the gel and the relaxation curves were averaged for each time
point. Analysis of force indentation curves for each set of target forces respectively yields
indentation depths of 1.99±0.02 μm, 2.77±0.03 μm and 3.86±0.06 μm and not
significantly different elastic moduli of 1.8±0.05 kPa, 1.9±0.06 kPa and 2±0.1 kPa.
Figure 4-2B shows the plot of force relaxation curves in log-log coordinates which
displays two clear plateaus at very short and at very long time scales for each set of
curves. The long timescale plateau indicates that the force relaxation in the gel has a clear
characteristic relaxation time. Furthermore these curves show that force relaxation in
these hydrogels does not exhibit a power law behaviour.
One hallmark of poroelastic materials is that their characteristic relaxation time
~ /p pt R Dd is dependent on the indentation depth d . In contrast, power law relaxations
of the form 0 0( ) ~ ( / )F t F t t b and exponential relaxations of the form
0( ) ~ exp( / )F t F t t have parameters b and t that are independent of length scale.
For ideal stress-relaxations of any power-law or linear viscoelastic material,
normalisation of force between any two arbitrarily chosen times 1t and 2t
1 2 1[ ( ) ( )]/[ ( ) ( )]F t F t F t F t should lead to collapse of all experimental relaxation
curves onto one master curve. However, for poroelastic materials, force normalisation
alone is not sufficient to collapse curves onto a single master curve and time must also be
renormalised with respect to the indentation depth to achieve collapse of curves.
100
Figure 4-2 Force-relaxation curves acquired with different indentation depths on hydrogel.
(A) Force-relaxation curves acquired on a hydrogel for three different indentation depths ( 1d , 2d and 3d ). Each
data point is the average of 20 measurements acquired at different positions on the hydrogel surface. (B) log-
log plot of the curves shown in (A). Curves show characteristic plateaus at short and long time-scales. In all
graphs, solid lines and grey shadings represent the mean and standard deviations at each data point,
respectively.
To distinguish between poroelastic and viscoelastic force-relaxations, I undertook a
rescaling analysis as explained in Chapter 3.8. First the force was normalised according to
equation (3.7) up to the time when each curve reaches its final plateau ( *t t ). As
shown in Figure 4-3A, normalisation of force alone did not result in the collapse of
curves onto a single master curve. If the gel were a linear viscoelastic material,
normalisation of force in this way should result in collapse of all curves onto a single
curve irrespective of indentation depth. However, if in addition to force, time is also
normalised by indentation depth, all experimental curves collapse onto a single curve,
indicating that poroelasticity is the dominant mechanism of force relaxation in hydrogels
(Figure 4-3B).
101
Figure 4-3 Normalisation of force-relaxation curves acquired on hydrogel.
(A) Normalisation of force-relaxation curves (Figure 4-2) using equation (3.7) plotted as a function of time.
For force renormalisation, a separate time *t was chosen for each curve such that *t t , with t the time
at which each curve reaches its plateau: 1 *t = 7.5 s, 2 *t = 10 s and 3 *t = 12.5 s. (B) Normalised force-
relaxation curves from A plotted as a function of time divided by indentation depth d ( /t d ). Time rescaling
was performed for each force relaxation curve separately using their respective indentation depth. Following
force renormalisation and time-rescaling, all force-relaxation curves collapsed onto one single curve. (C)
Normalisation of force-relaxation curves shown in Figure 4-2 using equation (3.7) plotted as a function
of time. For force renormalisation, a separate time *t dependent on indentation depth was chosen for each
curve with *t ad . Choosing a = 1.25 s.µm-1 gave times 1 *t = 2.5 s, 2 *t = 3.5 s and 3 *t = 4.9 s. (D)
Normalised force-relaxation curves from C plotted as a function of time divided by indentation depth d (
/t d ). Time rescaling was performed for each force-relaxation curve separately using their respective
indentation depth. Following force renormalisation and time-rescaling, all force-relaxation curves collapsed
onto one single curve. In all graphs, solid lines and grey shadings represent the mean and standard deviations
at each data point, respectively.
102
As explained earlier in Chapter 3.8, collapse of curves can also be obtained following
normalisation of time for times shorter than t . For times *t t to obtain collapse of
force relaxation of poroelastic material onto a master curve, *t must be selected
proportional to the indentation depth d specific to each curve: *t ad where a (with
unit of s.μm-1
) is an arbitrary number fixed for all curves (Figure 4-3C). With this choice
of *t a similar collapse of curves was observed using force and time normalisation
(Figure 4-3D), confirming the dominance of poroelasticity in the force relaxation of gels
in these experiments. Fitting the relaxation curves with equation (3.5) gave a poroelastic
diffusion constant of 33±4 μm.s-1
for the hydrogel comparable to the previously reported
values [137].
4.1.3 Cellular force-relaxation at short timescales is poroelastic
Population averaged force-relaxation curves showed similar trends for both HeLa and
MDCK cells with a rapid decay in the first 0.5 s followed by slower decay afterwards
(Figure 4-4A). When plotted in log-log scale in Figure 4-4B, we see that force-relaxation
clearly displayed two separate regimes: a plateau lasting ~ 0.1-0.2 s followed by a
transition to a linear regime. This indicated that at short time-scales cellular force-
relaxation did not follow a simple power law. Comparison with force-relaxation curves
acquired on physical hydrogels, which display a plateau at short time-scales followed by
a transition to a second plateau at longer time-scales (Figure 4-2A, B), suggests that the
initial plateau observed in cellular force-relaxation may correspond to poroelastic
behaviour. Indeed poroelastic models fitted the force-relaxation data well for short times
(< 0.5 s); whereas power law models were applicable for times longer than ~ 0.1-0.2 s
(Figure 4-4B).
103
Figure 4-4 Force-relaxation curves acquired with different indentation depths on cells.
(A) Population averaged force-relaxation curves for HeLa cells (green) and MDCK cells (blue) for target
indentation depths of 1.45 μm for HeLa cells and 1.75 μm for MDCK cells. Curves are averages of n = 5
HeLa cells and n = 20 MDCK cells. The grey shaded area around the average relaxation curves represents
the standard deviation of the data. (B) Population averaged force-relaxation curves for HeLa cells (green) and
MDCK cells (blue) from D-I plotted in a log-log scale. For both cell types, experimental force-relaxation was
fitted with poroelastic (black solid line) and power law relaxations (grey solid line).
To gain further insight into the nature of cellular force-relaxation at short time-scales, I
acquired experimental relaxation curves following indentations with increasing depths.
For analysis of these experiments, I selected cells with identical elasticities such that
application of a chosen target force resulted in identical indentation depths and I averaged
their relaxation curves. This filtering procedure ensured that the cellular relaxation curves
were comparable in all aspects (Figure 4-5A and Figure 4-6A) and allowed us to assess
whether or not cellular relaxation was dependent upon indentation depth. As shown
earlier for hydrogels, the most important property of these gels as a poroelastic material is
that their characteristic relaxation time depends on indentation depth. Indeed, the length
scale dependency of relaxation in poroelastic materials is in contrast to power law and
linear viscoelastic exponential relaxation behaviours.
104
Figure 4-5 Normalisation of force-relaxation curves acquired on MDCK cells.
(A) Force-relaxation curves acquired on MDCK cells for three different indentation depths ( 1d , 2d and 3d ).
Each relaxation curve is the average of 20 force-relaxation curves acquired on cells with matching elasticities
of E = 0.4±0.1 kPa. Elasticity matching ensured that identical forces were applied onto each cell to obtain a
given indentation depth. (B) Normalisation of force-relaxation curves shown in A using equation (3.7)
choosing *t = 3.6 s for all curves. Relaxation curves were significantly different from one another for times
shorter than ~ 0.5 s but appeared to collapse at times longer than ~ 1.2 s. (C) Normalisation of force-
relaxation curves shown in A using equation (3.7) plotted as a function of time. For force renormalisation, a
separate time *t dependent on indentation depth was chosen for each curve with *t ad . Choosing a =
0.16 s.µm-1 gave times 1 *t ~ 105 ms, 2 *t ~ 170 ms and 3 *t ~ 285 ms. (D) Normalised force-relaxation
curves from C plotted as a function of time divided by indentation depth d ( /t d ). Time rescaling was
performed for each force relaxation curve separately using their respective indentation depth d . Following
force renormalisation and time-rescaling, all three force-relaxation curves collapsed onto one single curve. In
all graphs, solid lines and grey shadings represent the mean and standard deviations at each data point,
respectively.
105
Figure 4-6 Normalisation of force-relaxation curves acquired on HeLa cells.
(A) Force-relaxation curves acquired on HeLa cells for two different indentation depths ( 1d and 2d ). Each
relaxation curve is the average of 5 force-relaxation curves acquired on cells with matching elasticities of E
= 0.66±0.1 kPa and heights of h = 5.5±0.3 μm. This ensured that identical forces were applied onto each cell
to obtain a given indentation depth. Matching of cell heights was necessary to minimise substrate effects due
to the smaller thickness of HeLa cells compared to MDCK cells. (B) Normalisation of force-relaxation curves
shown in A using equation (3.7) choosing *t = 1 s for all curves. Relaxation curves were significantly
different from one another for times shorter than ~ 0.5 s but appeared to collapse at times longer than ~ 0.6 s.
(C) Normalisation of force-relaxation curves shown in A using equation (3.7) plotted as a function of
time. For force renormalisation, a separate time *t dependent on indentation depth was chosen for each
curve with *t ad . Choosing a = 0.19 s.µm-1 gave times 1 *t ~ 170 ms and 2 *t ~ 275 ms. (D)
Normalised force-relaxation curves from C plotted as a function of time divided by indentation depth d (
/t d ). Time rescaling was performed for each force relaxation curve separately using their respective
indentation depth. Following force renormalisation and time-rescaling, both force-relaxation curves collapsed
onto one single curve. In all graphs, solid lines and grey shadings represent the mean and standard deviations
at each data point, respectively.
106
After force normalisation, experimental force-relaxation curves acquired on cells for
different indentation depths collapsed for times longer than ~ 1 s, but were significantly
different from one another for shorter times (Figure 4-5B for MDCK cells and
Figure 4-6B for HeLa cells), suggesting that, at short times, relaxation is length-scale
dependent consistent with a poroelastic behaviour.
For cells, experimental force-relaxation curves do not reach a plateau because relaxation
follows a power law at long time-scales (Figure 4-4B). Hence, to determine if relaxation
displayed a poroelastic behaviour at short time-scales, I followed the force normalisation
and time rescaling steps described in Chapter 3.8 for cases where *t t . These
procedures resulted in all cellular force-relaxation curves collapsing onto a single master
curve (Figure 4-5D for MDCK cells and Figure 4-6D for HeLa cells). This behaviour was
apparent in poroelastic hydrogels for both long and short time-scales (Figure 4-3B, D)
and in cells for times shorter than ~ 0.5 s (Figure 4-5D and Figure 4-6D). Taken together,
these data suggested that at short time-scales cellular relaxation is indentation depth
dependent and therefore that cells behaved as poroelastic materials.
4.2 Kinetics of whole cell swelling/shrinking
Using fluorescent particles attached to the cell membrane (see Figure 4-7A) and particle
tracking techniques, my aim here was to monitor the response of cells to osmotic
perturbations to investigate the swelling/shrinking kinetics of the cytoplasm. In particular
off-plane defocused images of fluorescent particles (depicted in Figure 3-6A, C) enabled
us to measure time-dependent cellular deformations in the z-direction very precisely.
HeLa cells with fluorescent beads bound to their membrane were suddenly exposed to a
change of osmolarity induced by addition of a small volume of water/sucrose (achieving
final concentrations of 50% water and 400 mM sucrose). A sudden change in osmolarity
induces osmotic pressure gradients that drive water in or out of the cell resulting in it
swelling or shrinking. After application of hypo/hyperosmotic shock, changes in the cell
107
height were measured over time by monitoring changes in the radius of the outer ring
(formed by the out of focus image of the fluorescent bead attached to the cell surface,
Figure 3-6A, B, C) to determine changes in z position of the bead.
Figure 4-7 Experimental setup to investigate the poroelastic behaviour of cells during swelling/shrinkage.
(A-I) Schematic of fluorescent microbeads attached to the cell membrane. (A-II) By changing the osmolarity
of the extracellular medium we can induce osmotic pressure gradients across the cell membrane that drive
water in or out of the cell resulting in swelling or shrinkage (Here a decrease in cell volume due to
hyperosmotic shock is depicted). Keeping the focal plane fixed, changes in the vertical position of each bead
due to swelling/shrinkage of the cell can be determined from the outer ring radius of the defocused image
using the calibration curve, Figure 3-6C. (B) Changes in height of beads over time after application of hypo-
osmotic/hyperosmotic shock at time 0t . Fitting the displacement curve to the analytical solutions for
poroelastic hydrogels allows the poroelastic properties of the cell to be estimated.
First, I asked whether the cell behaves as a poroelastic gel by fitting the experimental
displacement curves to the analytical solutions for swelling kinetics of poroelastic gels.
Then the poroelastic diffusion coefficient pD was computed and compared with the
values obtained from AFM indentation experiments in Chapter 5. As proposed in Chapter
6, this method could be employed to dissect the contribution of different cellular layers to
cell rheology and to determine the spatial distribution of pD .
108
4.2.1 Swelling/shrinkage of a constrained poroelastic material
Consider a thin layer of poroelastic material (a gel) attached on one side to a fixed
substrate and immersed in a solvent for sufficiently long to reach an equilibrium
homogeneous state with 0C and 0m the initial concentration and chemical potential of
the solvent respectively. When the gel is suddenly exposed to a solvent with different
chemical potential m , the chemical potential gradient 0m m drives the solvent to flow
into/out of the gel. The displacement and chemical potential fields can be considered one
dimensional in z if the lateral dimensions xL and yL of the gel are much larger than its
thickness H ( ,x yL L H ). Indeed, in this case, there is a negligible amount of flow
from the edges and the solvent flows mostly through top surface of the gel. The gel is
constrained in xy so 0xx yye e and the problem is similar to uniaxial strain
consolidation as described in Chapter 2 (2-4-2). For cells, constraint in xy is achieved
through focal contacts and integrin transmembrane proteins. The gel can freely swell and
shrink from the top surface ( 0zzs ) and thus the equations (2.48) and (2.50) can be
written in terms of chemical potential:
= 0
0
(1 2 )( )
2 (1 )(1 2 )
( ),(1 )
szzz
s s
sxx yy
s
ua
z G
b
n m me
nn m m
s sn
(4.1)
2
2.pDt x
m m (4.2)
Considering the top surface as the reference point, the spatial and temporal evolution of
chemical potential can be derived using appropriate boundary and initial conditions. The
equilibrium chemical potential before application of osmotic shock yields the initial
condition 0( , 0)zm m . A step change in chemical potential on the top surface which
remains constant over time (0, )tm m and the no flow condition at the bottom of the
109
gel | 0z Hzm
yield the boundary conditions. Therefore the solution of the diffusion
equation (4.2) is very similar to equation (2.52):
2 2
0 1,3,..
( , ) 4sin exp ,
2n
z t n zn
m Hm m p
p tm m p
(4.3)
where 2/4pD t Ht is the dimensionless time. One can also obtain the solution for
infinite thickness (also applicable for early times after shock) as:
0
( , ).
4 p
z t zerfc
D tm m
m m (4.4)
After a long enough time 2 /t H D , the solvent inside the gel has reached a new
equilibrium m m and the gel has shrunken/swollen from its initial height H , by h to
its final height. Considering the total final strain at very long times =zz
hH
e and
employing equation (4.1)-a, the final change in thickness of the gel is:
0(1 2 )
.2 (1 )
s
s s
h HG
n m mn
(4.5)
The change of thickness at top surface ( )h t is derived by integrating equation (4.1) and
using equations (4.3) and (4.5):
2 22 2
1,3,..
( ) 81 exp
n
h tn
h mp t
p (4.6)
Using equation (4.4)the change in the thickness of the gel is written as:
( ) 2
.pD th t
h H p (4.7)
110
Figure 4-8 Kinetics of cell swelling/shrinking in HeLa cells
(A, B) Swelling/shrinking kinetics of HeLa cells in response to hyperosmotic (A) and hypoosmotic (B)
shocks. Dynamics of cellular swelling/shrinking is precisely probed by tracking the height ( )h t of fluorescent
beads attached on the cell membrane at different heights H . The curves show the change cell height
normalised to the final change in thickness ( )/h t h . The inset shows the absolute change of height ( )h t . (C)
Swelling/shrinking curves were fitted to the poroelastic model using equation (4.6) derived for poroelastic
swelling of constrained gels. Fits to the equation (4.7) describes the early stage behaviour of cellular
swelling/shrinking.
111
4.2.2 Poroelasticity can explain the kinetics of cellular swelling/shrinking
During the experiments, HeLa cells were exposed to hyper/hypoosmotic medium at t =0
and the vertical displacement of the attached beads was measured as a function of time to
obtain swelling/shrinking curves (Figure 4-8A, B insets). These curves were then
normalized with respect to the final change in the height h (Figure 4-8A,B). The
( )/h t h curves were fitted with the analytical solution derived for a poroelastic
cytoplasm in equation (4.6) (Figure 4-8C). This solution fitted the experimental curves
well and fitting gave the poroelastic diffusion constants of pD =0.3±0.2 μm2.s
-1 (n = 16)
and pD =0.6±0.3μm2.s
-1 (n = 10)for swelling and shrinking respectively. Furthermore,
considering the early stage behaviour, the first 10 s of curves were fitted with analytical
solution in equation (4.7) and diffusion coefficients of pD = 0.4±0.2 μm2.s
-1 and pD =
0.7±0.3 μm2.s
-1 were obtained for swelling and shrinking respectively (Figure 4-8C). The
values obtained from fits with the short timescale solution were in good agreement with
those obtained fitting with the full analytical solution.
4.3 Discussion and conclusions
To validate the experimental strategy and analytical methods, first the force-relaxation
behaviour of hydrogels, which are well-characterised poroelastic materials, was
examined. Intrinsic viscoelasticity and poroelasticity concurrently contribute to the time
dependent deformation of gels. The intrinsic viscoelasticity is associated with
conformational changes in the solid meshwork, while poroelasticity is related to the
redistribution of fluid. During indentation tests on gels (similar to the hydrogel that is
used in my experiments) the viscoelastic relaxation time results from the occurrence of
multiple microscopic processes such as slippage and reptation of polymer chains [87].
This relaxation time is determined by the ratio of the shear viscosity h and the shear
modulus G : ~ /v Gt h
[87], and is indentation depth independent as both h and G are
112
independent of contact size. In contrast, the poroelastic relaxation time ~ /p pR Dt d
is
related to the time scale of fluid redistribution/migration within/out of the deformed
region and this depends on the contact radius and indentation depth. For hydrogels, plots
of force-relaxation curves in log-log coordinates suggest the involvement of a limited
number of relaxation processes consistent with poroelastic behaviour. More importantly,
for experiments performed with different indentation depths, appropriate normalisation of
experimental force-relaxation curves resulted in the collapse of all curves onto a single
master curve. This provides strong evidence for the dominance of poroelasticity in
relaxation of hydrogels in agreement with [121, 137, 142]. It may thus be concluded that
poroelasticity is the dominant mechanism of relaxation in our indentation experiments
on hydrogels.
A detailed examination of the functional form of cellular force-relaxation curves at
different time scales was undertaken and these curves were compared to force-relaxation
curves acquired on hydrogels. Force-relaxation induced by fast localised indentation
by AFM contained two regimes: at short time-scales, the time-scale of force- relaxation
was indentation depth dependent, a hallmark of poroelastic materials; while at
longer timescales relaxation exhibited a power law behaviour. This confirms that living
cells are much more complex active materials than simple hydrogels. Indeed,
at physiological time-scales (0.1-10s), in addition to conformational changes in polymeric
network of cytoskeleton and rearrangements in the solid phase (macromolecules) other
factors such as turnover of cytoskeletal fibres, activity of myosin molecular motors and
network rearrangements due to association and dissociation of crosslinkers, also influence
the relaxation significantly.
Cell mechanical studies over the years have revealed a rich phenomenological landscape
of rheological behaviours that are dependent upon probe geometry, loading protocol and
loading frequency [56, 58, 63-65], though the biological origin of many of these regimes
113
remains to be fully explored. It is widely recognized that over wide ranges of frequency,
the rheology of living cells exhibits a weak power law behaviour in which the viscoelastic
response does not exhibit any characteristic timescale (see Chapter 2.2.3). The structural
damping model, which has previously been applied to characterise a variety of biological
materials [143], was recently implemented to describe the observed power law behaviour
in cells [33]. In the power law structural damping model, the storage and loss moduli are
both weak power law functions of frequency: ' '', ~G G bw , 0 1b , and are related
through a structural damping coefficient '' '/ tan( /2)damp G Gh bp . As a result, in
this model, dissipation is linked to an elastic stress rather than an independent viscous
stress. A Newtonian viscous term m was added to the loss modulus of the model to
capture the observed dynamics at short timescales [112]. In magnetic twisting cytometry,
the addition of such a linear term could be to take into account the effects of drag forces
generated from viscous interaction between the probe and the surrounding fluid.
However, the calculated values for this viscous term ( 1m pa.s, [33, 79, 144]) are much
larger than the viscosity of surrounding fluid (~ 0.001 pa.s). Therefore, the physical
origins of this additional viscous parameter are still not clear and are therefore debatable.
The shape of my force-relaxation curves at short timescales was not consistent with a
power law behaviour (similar to experiments reported by others [53, 145-149]) and it is
surprising that the initial non-power law regime observed in my experiments should not
have been described before. I envisage two major reasons for this. First, most recent
experiments aim to describe the shear rheology of cells over many decades of frequency
and therefore apply oscillatory deformations to the cell, using shear rheology methods
rather than focusing on volumetric changes (dilatational rheology) as done here. Indeed,
measurements of shear rheology are of little or no consequence in understanding cell
rheology for which volumetric deformations are critical (e.g. cell migration [150] and the
formation of quasi-spherical protrusions known as blebs [107]). My experiments were
designed to apply large volumetric deformations via micro-indentation tests such that
114
they inherently capture the dilatational rheology of the cell. Indeed, my experiments are
placed in a regime where intracellular water flows should play an important role: rapid
application of a large volumetric deformation and observation of force relaxation over
short and intermediate time-scales using microindentation. Second, in experiments
reported by other groups, such as force-relaxation experiments similar to the ones
presented here [52, 53, 145, 151] and also using other very different experimental setups
[146, 147, 149], a rapid initial force relaxation was observed but its origin was not
examined. In these experiments, either the short timescale behaviours were neglected,
only the asymptotic part of relaxation was fitted, or functions such as double exponentials
with both fast and slow relaxation times or stretched exponential functions were used to
fit the experimental data.
To observe water flows in response to applied deformations, one needs to deform the cell
with a large probe over a short rise time rt . Indeed, the time scale of application of
deformation rt must be shorter than the time it takes for water to drain out of the
deformed volume 2 2 2~ / ~ /( )p pt L D L Em x (L is the probe length-scale and in the case
of indentation: ~L Rd with R the radius of the indenter and d indentation depth). In
the context of oscillatory sinusoidal loading, shorter rise time deformations are achieved
by increasing the frequency of excitation. However, if the loading frequency is too high,
water does not have time to drain out of the deformed volume before the next cycle of
deformation is applied. In this regime, water will not move relative to the mesh and
therefore inertial and structural viscous effects from the mesh are sufficient to describe
the system [81, 152]. Previous work has shown that for frequencies 2 2/( )pf E Lx m
such coupling occurs [81]. Hence, to observe poroelastic effects in oscillatory
experiments, it is required to deform the material at frequencies larger than 1/ pt to
satisfy the short rise time condition to induce fluid flow but smaller than pf so that fluid
and mesh are not coupled. As both conditions have identical scaling, it is expected that
they only be satisfied in a narrow window around pf ~ 5 Hz for my experimental
115
conditions. Most oscillatory loading experiments sample rheology in 2 points per decade
and hence would likely not detect this window. Hence, to observe poroelastic effects, we
need to apply deformations with rise times shorter than pt and repeated at frequencies
lower than pf .
It should be emphasised that this type of loading regime is very physiologically relevant.
Indeed, cells in tissues of the cardiovascular or respiratory systems are exposed to large
strain deformations (tens of %) applied at high strain rates (> 20%.s-1
) but with low repeat
frequencies (< 4 Hz) (e.g. 10% strain applied at ~ 50%.s-1
repeated at up to 4 Hz for
arterial walls [153], 70% strain applied at up to 900%.s-1
repeated at up to 4 Hz in the
myocardial wall [154], and 20% strain applied at > 20%.s-1
repeated at ~ 1 Hz for lung
alveola [155]). As the changes in cell shape that occur in physiological tissue deformation
are similar in magnitude or larger than those induced in our experiments and are repeated
with low frequency, I expect intracellular water redistribution to play an important role in
the relaxation of cells within these tissues
When force-relaxation curves acquired for different indentation depths on cells were
renormalized for force and rescaled with a time-scale dependent on indentation depth, all
experimental curves collapsed onto a single master curve for short time-scales,
confirming that the initial dynamics of cellular relaxation is dominated by poroelastic
effects and intracellular water flow (Figure 4-5 and Figure 4-6), but for larger timescales
the relaxation curves exhibit a weak power law behaviour. Together these data suggest
that in my experiments the power-law dynamics and poroelasticity coexist. For timescales
shorter than ~ 0.5 s, intracellular water redistribution contributed strongly to force-
relaxation, consistent with the ~ 0.1 s time-scale measured for intracellular water flows in
the cytoplasm of HeLa cells [156]. At long timescales when fluid is fully redistributed,
fluid related poroelastic dissipations became negligible and structural damping effects
(manifested in the form of a weak power law) come into play as the main source of
dissipation.
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Volume exclusion effects are the primary source of swelling/shrinking behaviour in
artificial and biological gels as well as living cells. The dynamics of swelling and
shrinking mainly depends on the rate at which water can be transported into and out of
the gel. However, poroelasticity can also significantly affect the swelling/shrinking
kinetics: for an elastic porous medium, the deformation of the porous matrix induced by
osmotic shock significantly limits the rate of swelling/shrinkage. Indeed a very stiff
sponge cannot swell/shrink even if water can diffuse into/out of it infinitely fast. Here the
hypothesis that the swelling/shrinking behaviour of cytoplasm is similar to that of
hydrogels was tested by examining its volume response to osmotic perturbations. It is
shown that that the swelling/shrinking kinetics of cytoplasm can be well described via
a simple 1-D poroelastic model and analytical solutions fitted the experimental data well.
The estimated values of the diffusion coefficient were within the previously reported
range of 0.1 to 10 μm.s-1 [100-102] but around one order of magnitude smaller than those
obtained from indentation experiments (as presented in next chapter). There are several
possible sources of this discrepancy as shall be explained in the following paragraph.
A one dimensional model for swelling/shrinking of a constrained gel was used to
determine the cellular poroelastic diffusion coefficient from fits of experimental curves.
This one dimensional formulation was a good approximation for my analysis because the
lateral movements of the membrane-attached beads were much smaller than their vertical
displacement (less than ~ 0.5 μm in xy versus greater than ~ 2 μm in z ). However this
model is only valid if the osmotic perturbations are applied in a step manner (infinitely
fast). In our experiments, it takes a certain amount of time for the added osmolyte to mix
with the bath solution and for osmolyte to equilibrate. This non instantaneous change in
osmolarity could be one reason for underestimation of diffusion coefficient using
proposed the poroelastic model. Comparing shrinking and swelling experiments, faster
poroelastic diffusion constants were obtained in shrinking experiments, which might be
due to irreversible dynamics of the membrane water channels (aquaporins) or different
117
responses of the solid phase and cytoskeleton to tensional versus compressional forces
created during hyperosmotic shrinkage and hypoosmotic swelling, respectively.
The structural heterogeneity and complex nature of the cell may be the key reason that
different experimental setups report different values for cellular mechanical and
rheological properties. AFM probes only small regions of cytoplasm whereas all of the
structures inside the whole cell are perturbed during a swelling/shrinking experiment.
Indeed several cellular layers, such as the cell membrane, the cell cortex, the nucleus, and
the granular filamentous interior, each with different permeabilities and elasticities,
contribute to setting the poroelastic behaviour of the cell in swelling/shrinking. During
indentation experiments, small regions of cytoplasm underneath the indentor are
compressed and water movement is directed towards the cell interior (see Figure 4-9). For
these indentation experiments, the cell membrane most likely does not play a significant
role in setting the rate of water permeation and the membrane can be assumed to be fully
permeable. In contrast, during swelling/shrinkage of the whole cell, water infiltration
through cytoplasm and all parts of the cell (including the nucleus) sets the dynamics of
swelling/shrinking. Specifically the rate of swelling/shrinking can be limited significantly
by the rate at which water can pass through thin layers of cytoplasm such as the cell
cortex and the plasma membrane.
118
Figure 4-9 Schematic representation of intracellular water movements in different experimental setups.
During osmotic shocks the water permeates through whole cytoplasm and all parts of the cell whereas for
AFM indentation tests water movements is restricted to small regions of cytoplasm underneath the indentor.
The arrows show possible directions of water movement.
The plasma membrane has long been thought of as the main regulator for volumetric
responses of cells to osmotic perturbations. However, it has been recently shown that in
the absence of a cell membrane the cytoplasm of mammalian cells has intrinsic
osmosensitivity [157]. Simple diffusion through the lipid bilayer and water selective
facilitated diffusion through the water channels (aquaporins) are the main pathways for
water permeation during osmotic perturbations [156, 158]. Coherent anti-Stokes Raman
scattering (CARS) microscopy allows observation of intracellular hydrodynamics by
monitoring H2O/D2O exchange in living cells [159]. Using CARS, recent experimental
measurements found that for Hela S3 cells it takes less than ~ 100 ms for water to fully
permeate across the plasma membrane and fully exchange, much shorter than our
measured relaxation times (>> 1 s) for the volumetric response of HeLa cells to osmotic
perturbations. Therefore, in addition to diffusional effects, mechanical factors also play
an important role in setting the relaxation time. Furthermore, considering a cell height of
~ 4 μm and a poroelastic diffusion coefficient ~ 40 μm2.s
-1 yields the ratio of /pD h ~ 10
μm.s-1
which is of the magnitude as the measured membrane diffusional permeability ~
10-40 μm.s-1
[156, 159] suggesting that the solid phase of cytoplasm has a similar
119
contribution in limiting the rate of water permeation to the membrane. I conclude that the
simple concept of a water-filled container surrounded by membrane, cannot fully explain
the volumetric response of cell to osmotic perturbations and that the solid phase of
cytoplasm contributes significantly in this response. I suggest that poroelastic model for
the cytoplasm is a more physical concept that inherently considers the gel-like nature of
cytoplasm and can help explain the dynamics of volume changes. However, further
experiments suggested in Chapter 6 are required to precisely dissect the role of the
membrane and different parts of the solid phase in determining the cell’s response to
osmotic shocks.
120
Chapter 5
Poroelastic properties of
cytoplasm: Osmotic perturbations
and the role of cytoskeleton
Equation Chapter (Next) Section 1
It has been shown in the previous chapter that the framework of poroelasticity is well
suited to understanding the time-dependent mechanical properties of cells. In this
framework, the cytoplasm comprises two phases: a porous solid phase (cytoskeleton,
organelles, macromolecules) and a fluid phase (cytosol). The conceptual advantage of
such a description is that cellular rheology can be related to measurable cellular
parameters (elastic modulus E , hydraulic pore size x , and cytosolic viscosity m ) using a
simple scaling law 2~ /pD Ex m and therefore changes in rheology resulting from
changes in cellular organisation can be qualitatively predicted. In this chapter, AFM
indentation tests were utilized to measure cellular stress-relaxation in response to rapid
121
local force application in conjunction with chemical, genetic, and osmotic treatments to
modulate two of the parameters that scaling law predicts should influence cellular time-
dependent mechanical properties: pore size x and the elasticity E .
5.1 Determination of cellular elastic, viscoelastic and poroelastic
parameters from AFM measurements
To provide baseline behaviour for perturbation experiments, I measured the elastic,
viscoelastic and poroelastic properties of MDCK, HeLa, and HT1080 cells by fitting
force-indentation (Figure 3-3) and force-relaxation (Figure 5-1) curves with Hertzian and
viscoelastic or poroelastic models, respectively. Measurement of average cell thickness
suggested that for time-scales shorter than 0.5 s, cells could be considered semi-infinite
and forces relaxed according to an exponential relationship with 2( ) exp( / )pF t D t L
(see Chapter 2.4, 5, Figure 2-5 and Figure 2-7C). In our experimental conditions (3.5-6
nN force resulting in indentation depths less than 25% of cell height, Figure 4-1B, C),
force-relaxation with an average amplitude of 40% was observed with 80% of total
relaxation occurring in ~ 0.5 s (Figure 4-4 and Figure 5-1A).
Analysis of the indentation curves yielded an elastic modulus of E = 0.9±0.4 kPa for
HeLa cells (N = 189 curves on n = 25 cells), E = 0.4±0.2 kPa for HT1080 cells (N =
161 curves on n =27 cells), and E = 0.4±0.1 kPa for MDCK cells (n = 20 cells).
Poroelastic models fitted experimental force-relaxation curves well (on average 2r =
0.95, black line, Figure 5-1A) and yielded a poroelastic diffusion constant of pD = 41±11
µm2.s
-1 for HeLa cells, pD = 40±10 µm
2.s
-1 for HT1080 cells, and pD = 61±10 µm
2.s
-1 for
MDCK cells.
122
Figure 5-1 Fitting force-relaxation curves.
(A) The first 0.5 s of force-relaxation curves were fitted with the poroelastic model (black line) and the
standard linear viscoelastic model (grey line). Poroelastic models fitted the data better than the single phase
viscoelastic model especially at short times. Inset: the percentage error of each fit defined as | |
. (B) Ratio of root mean square error (RMSE) of the viscoelastic fit to the RMSE of poroelastic fit.
Assuming that the cytoplasm behaves as a standard linear viscoelastic material and to
measure the apparent viscosity of the cytoplasm, I fitted force-relaxation curves with
equation (3.6) ( 1( ) ~ exp( / )F t k t h ). Using the approach phase of AFM indentation
curves, I measured cellular elasticity and assumed 1 ~k E . Therefore the relaxations
predicted by a viscoelastic formulation depend on only one free parameter: the apparent
viscosity h . Since the force-relaxation curves are indentation depth dependent, to
calculate the apparent viscosity I considered only the curves with indentation depths of
±10% of the mean values obtained from distribution of indentation depths (Figure 4-1B).
Viscoelastic formulations were found to replicate experimental force-relaxation curves
well (on average 2r = 0.92), but the early phase of force-relaxation was fit somewhat less
accurately by viscoelastic models (gray line, Figure 5-1A) compared to poroelastic fits
(black line, Figure 5-1A). Indeed, viscoelastic models gave rise to somewhat larger errors
123
than poroelastic models (Figure 5-1B). An average apparent viscosity of h = 166±81 Pa.s
was measured for HeLa cells, h = 77±37 Pa.s for HT1080 cells and h = 50±15 Pa.s for
MDCK cells.
5.2 Poroelasticity can predict changes in stress relaxation in
response to volume changes
Having established that poroelastic models provide good fits for stress relaxation in cells,
I examined the ability of our simple scaling law to qualitatively predict changes in pD
resulting from changes in pore size due to cell volume change. Indeed, cell volume
changes should not affect cytoskeletal organisation or integrity but should alter
cytoplasmic pore size. In our experiments, we exposed Hela and MDCK cells to media of
different osmolarities: hyperosmotic media to decrease cell volume and hypoosmotic
media to increase cell volume.
First, I measured cell volume change in response to changes in osmolarity for up to 30
min to ascertain that volume changes persisted long enough for experimental
measurements to be carried out with AFM (Figure 5-2A). Indeed, cell volume stayed
constant over this time frame. I verified that photobleaching due to the acquisition of
confocal stacks over an extended period of time did not artefactually affect our
measurement of cell volume (control, n = 6 cells, Figure 5-2A). Cells regulate their
volume tightly to be able to tolerate extracellular environments of different osmolarities.
When osmolarity is perturbed, flow of water across cell membrane causes cell swelling or
shrinkage. However, the cell can restore its original volume by activation of channels or
transporters (typically K+, Cl and organic osmolytes) to lose or gain sufficient amount of
osmolytes to return to its preferred volume. To ensure a stable volume increase under
hypoosmotic conditions over the timescale necessary for AFM experiments (~ 30 min),
cells were treated simultaneously with regulatory volume decrease (RVD) inhibitors
124
[160]. Also, MDCK cells were treated with regulatory volume increase (RVI)
inhibitors to ensure a stable decrease in cell volume after shrinkage. Under these
conditions, a stable increase of 22±2% in cell volume was measured after hypoosmotic
treatment (n = 3 cells, Figure 5-2A). Conversely, upon addition of 240 mM sucrose, cell
volume decreased by 21±6% (n = 10 cells, Figure 5-2A) and upon addition of PEG-400
(30% volumetric concentration), cell volume decreased drastically to 46±3% of the
original volume (N = 6 cells, Figure 5-2A)†.
Next, using AFM indentations I investigated whether changes in cell volume resulted in
changes in the poroelastic diffusion constant. Increases in cell volume resulted in a
significant increase (55%) in poroelastic diffusion constant pD and in a significant
decrease (-20%) in cellular elasticity E (N = 92 measurements on n = 33 cells, p <
0.01 when compared to control for both E and pD , Figure 5-2B). In contrast, cells
exposed to PEG-400 exhibited a two-fold lower diffusion constant ( p < 0.01 when
compared to controls) than control cells and a 50% larger elasticity ( p < 0.01, N = 176
measurements on n = 17 cells) (Figure 5-2B). Addition of sucrose with low
concentrations (<<250 mM) resulted in no significant changes in elasticity or diffusion
constant ( p = 0.53 for E and p = 0.89 for pD ).
As will be examined in detail in next section, there are spatial inhomogeneities in shape
architecture and mechanical properties of HeLa cells spread on glass substrates. To
increase the accuracy of the measurements and decrease the uncertainties arising from
spatial inhomogeneities, similar experiments were performed on MDCK cells with 8 data
points corresponding to sampling different medium osmolarities (Figure 5-2C).
Qualitatively, both cell types had the same behaviour, confirming the generality of our
results (Figure 5-2C, D).
† I would like to acknowledge the preliminary data from report of L. Valon.
125
Figure 5-2 Poroelastic and elastic properties change in response to changes in cell volume.
In all graphs, error bars indicate the standard deviation and, in graphs B and C, asterisks indicate significant
changes compared to control ( p < 0.01). N indicates the total number of measurements and n indicates the
number of cells. In hypoosmotic shock experiments, cells were incubated with NPPB and DCPIB (N+D on
the graph), inhibitors of RVD. In hyperosmotic shock experiments, MDCK cells were incubated with
RVI inhibitors (EIPA). (A) Cell volume change over time in response to changes in extracellular osmolarity.
The volume was normalized to the initial cell volume at t = 0 s. The arrow indicates the time of addition of
osmolytes. (B) Effect of osmotic treatments on the elasticity E (squares) and poroelastic diffusion constant
pD (circles) in HeLa cells. (C) Effect of osmotic treatments on the elasticity E (squares) and poroelastic
diffusion constant pD (circles) in MDCK cells. (D) pD and E plotted as a function of the normalised
volumetric pore size 1/30~ ( / )V Vy in log-log plots for MDCK cells (black squares and circles) and HeLa
cells (grey squares and circles). Straight lines were fitted to the experimental data points weighted by the
number of measurements to reveal the scaling of pD and E with changes in volumetric pore size (grey
lines for HeLa cells, 1.6~E y and 2.9~pD y and black lines for MDCK cells, 5.9~E y and 1.9~pD y ).
126
Consistent with results reported by others [125], our experiments revealed that cells
relaxed less rapidly and became stiffer with decreasing fluid fraction. Increases in cell
volume resulted in a significant increase in poroelastic diffusion constant pD and a
significant decrease in cellular elasticity E (Figure 5-2B, C). In contrast, a decrease in
cell volume decreased the diffusion constant and increased elasticity (Figure 5-2B, C).
Because the exact relationship between hydraulic pore size x and cell volume is
unknown, pD and E were plotted as a function of the change in the volumetric pore size
1/30~ ( / )V Vy . As shown in the log-log plot in Figure 5-2D, for both MDCK and HeLa
cells, pD scaled with y and the cellular elasticity E scaled inversely with y . In
summary, an increase in cell volume increased the poroelastic diffusion constant and
decrease in cell volume decreased pD qualitatively consistent with our simple scaling
law.
To exclude any possible indirect effect of volume change due to changes in cytoskeletal
organisation, I verified that cytoskeletal structure was not perturbed by changes in cell
volume (Figure 5-3 for actin).
127
Figure 5-3 Effects of hypoosmotic or hyperosmotic shock on HeLa cells F-actin structural organization.
The F-actin cytoskeleton does not reorganise in response to hypoosmotic treatment (A) or hyperosmotic
treatment (B). All images are maximum projections of a confocal image stack of cells expressing Life-act
ruby (red). Nuclei were stained with Hoechst 34332 (blue). In A and B, image (I) shows the cell before
addition of osmolyte and image (II) after addition. Scale bar = 10 µm. No dramatic changes in structure of
actin filaments could be observed after hypoosmotic (A) or hyperosmotic (B) shock.
5.3 Changes in cell volume result in changes in cytoplasmic pore
size.
Having shown that changes in cell volume result in changes in the poroelastic diffusion
constant without affecting cytoskeletal structure, I tried to directly detect if the volume
perturbations resulted in changes of pore size. As shall be described in the following,
experiments examining the mobility of microinjected quantum dots (qdots) and
photobleaching experiments were performed to show that osmotic perturbations and the
concomitant volume changes directly affect the cytoplasmic pore size. Furthermore, as
poroelasticity is related to fluid redistribution phenomena, I verified the existence of a
128
fluid phase in cells under extreme shrinkage conditions by photobleaching of small
cytoplasmic fluorescein analogs.
5.3.1 Hyperosmotic shock halts movement of quantum dots inside cytoplasm
PEG-passivated qdots (nanoparticles made of a semiconductor material with unique
optical and electrical properties, ~ 14 nm hydrodynamic radius with the PEG-passivation
layer [161]) were microinjected into cells and their mobility was examined before and
after application of hyperosmotic shock. Under isoosmotic condition, qdots rapidly
diffused throughout the cell; however, upon addition of PEG-400, they became immobile
(supplementary video 1, Figure 5-5A-I, n = 7 cells examined). Hence, cytoplasmic pore
size decreased in response to cell volume decrease trapping qdots in the cytoplasmic solid
fraction and immobilising them (Figure 5-5A-II). This suggested that the isoosmotic pore
radius x was larger than 14 nm, consistent with our estimates from poroelasticity (see
section 5.5).
129
Figure 5-4 Fluorescence recovery after photobleaching in HeLa cells.
(A) Fluorescence recovery after photobleaching of monomeric free diffusing GFP in isosmotic conditions.
Solid lines indicate fluorescence recovery after photobleaching and are the average of N = 18 measurements
on n = 7 cells. The experimental recovery was fitted with the theoretical model described in Chapter 3.11
(grey line). (B) Fluorescence recovery after photobleaching of GFP-actin in the cortex of HeLa cells blocked
in prometaphase. Solid lines indicate fluorescence recovery after photobleaching and are the average of N =
10 measurements on n = 4 cells. A half time recovery of 1/2, F actint = 11.2±3 s was obtained.In (A) and (B),
dashed lines indicate loss of fluorescence due to imaging in a region outside of the zone of FRAP, error bars
indicate the standard deviation for each time point, and the greyed area indicates the duration of
photobleaching.
5.3.2 Translational diffusion of fluorescent probes affected by osmotic
perturbations
5.3.2.1 Effective bleach radius in FRAP experiments.
To derive quantitative measurements of the translational diffusion constants of molecules
within the cytoplasm using the methods described in [141], I experimentally determined
the effective bleach radius in the first post-bleach image for a nominal bleaching zone of
diameter 0.5 µm (Figure 3-7III). Averaging over all diameters passing through the centre
130
of the nominal bleach regions yielded an experimental curve that could be fit with
equation (3.11) to find er = 1.5±0.2 µm (S5A-V, N = 6 measurements on n = 6 cells).
5.3.2.2 Measurement of translational diffusion constants in cells.
To validate my estimations of the translational diffusion constant TD using the analytical
procedures described in Chapter 3.11, I first measured TD for GFP molecules and small
fluorescein analogs (CMFDA, hydrodynamic radius hR ~ 0.9 nm [162]) freely diffusing
in the cytoplasm. For cytoplasmic GFP molecules, I found ,T GFPD ~ 25 µm2.s
-1
(Figure 5-4A, N = 18 measurements on n = 7 cells), consistent with values reported by
others [141]. I also estimated a translational diffusion constant ,T CMFDAD ~ 38 µm2.s
-1 for
CMFDA (N = 19 measurements on n = 7 cells) which is in a good agreement with
published values in cells (Figure 5-5B, TD = 24-40 µm2.s
-1, [162]). Indeed, effects of
molecular crowding and hindered diffusion inside the cell slow the diffusion of GFP and
CMFDA by more than ~3 fold compared to diffusion of these analogs in aqueous
solutions ( , ,T GFP aqueousD ~ 90 µm2.s
-1 and , ,T CMFDA aqueousD ~ 240 µm
2.s
-1 [162, 163]). It
should be noted that with CMFDA loss of fluorescence due to imaging is very high (also
reported by others, Figure 5-5B). Therefore, only data points up to a total fluorescence
loss of 15% due to imaging were utilised to estimate the translational diffusion of
fluorophores in cells in normal isotonic condition.
5.3.2.3 Osmotic perturbations change the cytoplasmic pore size
In isoosmotic conditions, CMFDA recovered rapidly after photobleaching (black line,
Figure 5-5B, Table 5-1) whereas recovery slowed three fold in the presence of PEG-400
consistent with [164] (grey line, Figure 5-5B and Table 5-1). Recovery after
photobleaching of CMFDA under strong hyperosmotic conditions confirmed the presence
of a fluid-phase indicating that under severe hyperosmotic conditions cells still retained a
significant fluid fraction. To estimate the reduction in translational diffusion of CMFDA
131
in cells under hyperosmotic conditions, I normalised the data following the procedures
described in [165] and this gave an estimated value of , ,T CMFDA HyperD ~ 14 µm2.s
-1 (N =
20 measurements on n = 5 cells). The measured decrease in translational diffusion
suggested a reduction in the cytoplasmic pore size with dehydration. Indeed, translational
diffusion is related to the solid fraction f via the empirical relation / ~ exp( )T TD D f
with TD the translational diffusion constant of the molecule in a dilute isotropic
solution [166]. Assuming that the fluid is contained in N pores of equal radius x , the
solid fraction is 3~ /( )s sV V Nf x with sV the volume of the solid fraction, a constant.
/T TD D is therefore a monotonic increasing function of x . Therefore the decrease in
TD measured in response to hyperosmotic shock suggests a decrease in x . To examine
the effect of volume increase on pore size, I examined the fluorescence recovery after
photobleaching of a cytoplasmic GFP decamer (EGFP-10x, hR ~ 7.5 nm). Increases in
cell volume resulted in a significant ~ 2-fold increase in TD ( p < 0.01, Figure 5-5C,
Table 5-1), suggesting that pore size did increase. Together, our experiments show that
changes in cell volume modulate cytoplasmic pore size x consistent with estimates from
AFM measurements (Figure 5-11).
Table 5-1 Effects of osmotic perturbations on translational diffusion coefficients.
Translation diffusional coefficients TD are reported with unit of (µm2.s-1).
GFP
Control
N = 18, n = 7
CMFDA
Control
N = 19, n = 7
CMFDA
PEG-400
N = 20, n = 5
EGFP10x
Control
N = 17, n = 6
EGFP10x
Water+N+D
N = 23, n = 7
24.9±7 38.3±8 13.7±4 8.6±2 15.5±5
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Figure 5-5 Changes in cell volume change cytoplasmic pore size.
(A) Movement of PEG-passivated quantum dots microinjected into HeLa cells in isoosmotic conditions (I)
and in hyperosmotic conditions (II). Both images are a projection of 120 frames totalling 18 s (Supplementary
movie). In isoosmotic conditions, quantum dots moved freely and the time-projection appeared blurry (I);
whereas in hyperosmotic conditions, quantum dots were immobile and the time-projected image allowed
individual quantum dots to be identified (II). Images A-I and II are single confocal sections. In (B) and (C),
dashed lines indicate loss of fluorescence due to imaging in a region outside of the zone where fluorescence
recovery after photobleaching (FRAP) was measured, solid lines indicate fluorescence recovery after
photobleaching and are the average of N measurements and error bars indicate the standard deviation for
each time point. The greyed area indicates the duration of photobleaching. (B) FRAP of CMFDA (a
fluorescein analog) in isoosmotic (black, N = 19 measurements on n = 7 cells) and hyperosmotic
conditions (grey, N = 20 measurements on n = 5 cells). In both conditions, fluorescence recovered after
photobleaching but the rate of recovery was decreased significantly in hyperosmotic conditions. (C) FRAP of
EGFP-10x (a GFP decamer) in isoosmotic (black, N = 17 measurements on n = 6 cells) and hypoosmotic
conditions (grey, N = 23 measurements on n = 7 cells). The rate of recovery was increased significantly in
hypoosmotic conditions.
133
5.4 Estimation of the hydraulic pore size from experimental
measurements of Dp.
To estimate HeLa cells hydraulic pore size, equation (2.38), 2 /p sD Ea x m , can be
used. As a first approximation, I considered sn ~ 0.3 and a fluid fraction of j ~ 0.75 for
cells in isotonic conditions which yields a Kozney constant of k ~ 4 assuming the lower
bound for a random distribution of particulate spheres [167]. These values yield an
estimate of ~ 0.05 for a in isotonic conditions. Multiple experimental reports have
shown that for small molecules the viscosity m (~ 0.001 Pa.s) of the fluid-phase of
cytoplasm (cytosol) is 2-3 fold higher in cells than in aqueous media [71, 168]. Using
2 /p sD Ea x m , together with our experimental measurements of pD ~ 40 µm2.s
-1 and
sE ~ 1 kPa in HeLa cells yields a hydraulic pore size x of ~ 15 nm, comparable to the
value estimated in section 5.3.
5.5 Spatial variations in poroelastic properties
Maps of cellular elasticity measured by AFM show that E is strongly dependent upon
location within the cell with actin-rich organelles (lamellipodium, actin stress fibres)
appearing significantly stiffer than other parts of the cell [169]. Hence, I asked if
something similar could be observed for poroelastic properties by measuring the cellular
poroelastic properties at different locations along the cell long axis (Figure 3-5). I
performed N = 386 total measurements on n = 30 cells and displayed the measurements
averaged over 2 µm bins as a function of distance to the centroid of the nucleus. Cell
height decreased significantly with increasing distance from the nucleus (Figure 5-6A)
and this measurement enabled us to exclude low areas of the cell that are prone to
measurement artefacts due to the limited thickness. The poroelastic diffusion constant pD
decreased slightly, but not significantly, away from the nucleus going from 40 µm2.s
-1 to
134
~ 30 µm2.s
-1 ( p > 0.02, Figure 5-6C). In contrast, elasticity increased significantly away
from the nucleus increasing from ~ 500 Pa to ~ 1200 Pa in lower areas (Figure 5-6B).
Finally, the lumped pore size estimated from the ratio 1/2/pD Em also decreased
significantly away from the nucleus (Figure 5-6D).
Figure 5-6 Spatial distribution of cellular rheological properties.
(A-D) Scatter plots of the average cell height h , elasticity E , poroelastic diffusion constant pD , and
lumped pore size 1/2
/pD Em as a function of distance to the centre of nucleus. Each scatter graph is
generated by averaging values of each variable over 2 μm wide bins. Filled black circles indicate locations
above the nucleus, white unfilled circles in the cytoplasm, and grey filled circles boundary areas. The lines
indicate the weighted least-square fit of the scattered data. Unfilled circles with a cross superimposed indicate
measurements significantly different ( p < 0.01) from those obtained above the nucleus.
135
5.6 Poroelastic properties are influenced by cytoskeletal integrity
It has been shown previously that cellular poroelastic properties are affected by the cell
cycle state of the cell [101]. Specifically, in mitotic cells blocked in metaphase ( F-actin
and vimentin intermediate filaments concentrate at the cell periphery, microtubules
reorganize to form the mitotic spindle and cytokeratin filaments disassemble) a ~ two fold
larger poroelastic diffusion constant was measured compared to interphase cells [101]. It
has been hypothesized that this may be due to the dramatic reorganization of the
cytoskeleton concomitant with metaphase [101]. Cytoskeletal organisation strongly
influences cellular elasticity E [169-171], and is also likely to affect the cytoplasmic
pore size x (Figure 5-7A-C). As both factors (elasticity and pore size) play antagonistic
roles in setting pD , in the following I examined the effect of disruption of the
cytoskeleton on the cellular poroelastic properties using chemical and genetic treatments.
5.6.1 Chemical treatments
Treatment of cells with 750 nM latrunculin, a drug that depolymerises the actin
cytoskeleton, resulted in a significant decrease in the cellular elastic modulus
(Figure 5-9A, consistent with [169]), a ~ two-fold increase in the poroelastic diffusion
coefficient (Figure 5-9B), and a significant increase in the lumped pore size
(Figure 5-11C). Depolymerisation of microtubules by treatment with 5 µM nocodazole
had no significant effect (Figure 5-9 and Figure 5-11C). Stabilisation of microtubules
with 350 nM taxol resulted in a small (-16%) but significant decrease in elasticity but did
not alter pD (Figure 5-9 and Figure 5-11C). Perturbation of contractility with the myosin
II ATPase inhibitor blebbistatin (100 μM) lead to a 50% increase in pD , a 70% decrease
in E , and a significant increase in lumped pore size (Figure 5-9 and Figure 5-11C)‡.
‡ I would like to acknowledge the preliminary data from report of L. Valon.
136
Figure 5-7 Cellular distribution of cytoskeletal fibres.
Scale bars = 10 µm. Nuclei are shown in blue and GFP-tagged cytoskeletal fibres are shown in green. (A)
Maximum projection of a cell expressing tubulin-GFP. GFP-α-tubulin was homogenously localised
throughout the cell body. (B) Maximum projection of cells expressing Life-act-GFP, an F-actin reporter
construct. F-actin was enriched in cell protrusions but was also present in the cell body. (C) Maximum
projection of cells expressing keratin18-GFP. Intermediate filaments (keratin 18) were homogenously
distributed throughout the cell body. (D) Overexpression of the dominant mutant Keratin 14 R125C resulted
in aggregation of the cellular keratin network. (A&B were acquired by L. Valon).
5.6.2 Genetic modification
In light of the dramatic effect of F-actin depolymerisation on pD and x , we attempted to
decrease the pore size by expressing a constitutively active mutant of WASp (WASp
I294T, CA-WASp) that results in excessive polymerization of F-actin in the cytoplasm
through ectopic activation of the arp2/3 complex, an F-actin nucleator (Figure 5-8, [26]).
Increased cytoplasmic F-actin due to CA-WASp resulted in a significant decrease in the
137
poroelastic diffusion coefficient (-43%, p < 0.01, Figure 5-9B), a significant increase in
cellular elasticity (+33%, p < 0.01, Figure 5-9A), and a significant decrease in the
lumped pore size (-38%, p < 0.01, Figure 5-11C). Similar results were also obtained for
HT1080 cells (-49% for pD , +68% for E and -38% for lumped pore size, p < 0.01).
Figure 5-8 Ectopic polymerization of F-actin due to CA-WASp.
HeLa cells were transduced with a lentivirus encoding GFP-CA-WASp (in green) and stained for F-actin with
Rhodamine-Phalloidin (in red). Cells expressing high levels of CA-WASp (green, I) had more cytoplasmic F-
actin (red, II, green arrow) than cells expressing no CA-WASp (II, white arrow). (III) zx profile of the cells
shown in (I) and (II) taken along the dashed line. Cells expressing CA-WASp displayed more intense
cytoplasmic F-actin staining (green arrow) than control cells (white arrow). Cortical actin fluorescence levels
appeared unchanged. Nuclei are shown in blue. Images I and II are single confocal sections acquired with
kind helps from Dr. Moulding. Scale bars = 10 µm.
138
Figure 5-9 Effect of drug treatments and genetic perturbations on poroelastic properties.
Effect of F-actin depolymerisation (Latrunculin treatment), F-actin overpolymerisation (overexpression of
CA-WASp), myosin inhibition (blebbistatin treatment), microtubule stabilisation (Taxol treatment),
microtubule over-polymerisation (overexpression of γ-tubulin), microtubule depolymerisation (nocodazole
treatment), intermediate filaments aggregation (expression of Keratin 14 R125C) and crosslinking
perturbation (overexpression of ΔABD-α-actinin) on the cellular elasticity E in (A) and poroelastic
diffusion constant pD in (B). Asterisks indicate significant changes ( p < 0.01). N is the total number of
measurements and n the number of cells examined.
139
Then, we attempted to change cell rheology without affecting intracellular F-actin
concentration by perturbing crosslinking or contractility. To perturb crosslinking, we
overexpressed a deletion mutant of α-actinin (ΔABD-α-actinin) that lacks an actin-
binding domain but can still dimerize with endogenous protein [172], reasoning that this
should either increase the F-actin gel entanglement length l or reduce the average
diameter of F-actin bundles b (the inverse scenario is depicted in Figure 5-12B).
Overexpression of ΔABD-α-actinin led to a significant decrease in E but no change in
pD or lumped pore size (Figure 5-9 and Figure 5-11C). To determine if ectopic
polymerisation of microtubules had similar effects to CA-WASp, we overexpressed γ-
tubulin, a microtubule nucleator [173], but found that this had no effect on cellular
elasticity, poroelastic diffusion constant, or lumped pore size (Figure 5-9 and
Figure 5-11C). Finally, expression of a dominant keratin mutant (Keratin 14 R125C,
[130]) that causes aggregation of the cellular keratin intermediate filament network had
no effect on E , pD , or lumped pore size (Figure 5-9 and Figure 5-11C).
5.6.2.1 Clinical relevance: Mechanical disruption of mitosis in cells with
dysregulated F-actin production:
Dynamic events during cell division, such as the poleward movement of chromosomes or
the closure of the cleavage furrow, are mechanical processes that are intrinsically
linked to the physical properties of the cell. It is well established that actin is the main
cytoskeletal determinant of cellular rheology (as in this thesis under framework of
poroelasticity), and provides the physical support to maintain cell shape and drive shape
change [169-171]. In some patients, malfunctioning of the WASp protein can lead to
severe immunodeficiency, known as x-linked neutropenia. CA-WASp (a constitutive
WASp activation) causes increased F-actin production through dysregulated activation of
the Arp2/3 complex throughout the cytoplasm, which results in loss of proliferation, high
levels of apoptosis, cell division defects and chromosomal instability [26, 27]. In a very
recent study, we investigated the role of WASp in cell division and asked if an increased
140
abundance of F-actin in CA-WASp could cause these defects (Moulding et al, Blood,
2012). We showed that excessive cytoplasmic F-actin increases the elasticity and
apparent viscosity of the cytoplasm and mechanically impedes chromosome separation
and furrow closure during mitosis. Inhibition of the Arp2/3 complex reduced the change
in viscosity, restored the velocity of chromosome movement and reversed the XLN
phenotype. Taken together, these results show that non-specific alterations to cellular
rheology can lead to disease by perturbing critical functions such as mitosis.
Figure 5-10 Cell rheology and disease.
AFM measurements on HT1080 cells revealed a ~ two-fold increase in both cellular elasticity and apparent
cellular viscosity due to CA-WASp expression. Inhibition of the Arp2/3 complex using different
concentration of the CK666 drug reduced the amount of cytoplasmic F-actin in CA-WASp cells, leading to
partial recovery of the cell rheological properties and reducing cell division defects (Moulding et al, Blood,
2012)
141
5.7 Discussion and conclusions
I have shown that water redistribution plays a significant role in cellular responses to
mechanical stresses and that the effect of osmotic and cytoskeletal perturbations on
cellular rheology can be understood in the framework of poroelasticity through a simple
scaling law 2~ /pD Ex m . I tested the dependence of pD on the hydraulic pore size x by
modulating cell volume and showed that pD scaled proportionally to volume change,
consistent with a poroelastic scaling law. Changes in cell volume did not affect
cytoskeletal organisation but did modulate pore size. Experiments monitoring the
mobility of microinjected quantum dots suggested that x was ~ 14 nm consistent with
estimates based on measured values for pD and E . I also confirmed that cellular
elasticity scaled inversely with volume change as shown experimentally [125, 174] and
theoretically [84] for F-actin gels and cells. However, the exact relationship between the
poroelastic diffusion constant pD and the hydraulic pore size x could not be tested
experimentally because the relationship between volumetric change and change in x is
unknown due to the complex nature of the solid phase of the cytoplasm (composed of the
cytoskeletal gel, organelles, and macromolecules, Figure 5-12A, [175]). Taken together,
the results show that, for time-scales up to ~ 0.5 s, the dynamics of cellular force-
relaxation is consistent with a poroelastic behaviour for cells and that changes in cellular
volume resulted in changes in pD due to changes in x .
Although the cytoskeleton plays a fundamental role in modulating cellular elasticity and
rheology, our studies (it should be note that the indentation method that is used here only
probes local rheological properties of the cell) show that microtubules and keratin
intermediate filaments do not play a significant role in setting cellular rheological
properties (Figure 5-9 and Figure 5-11C). In contrast, both the poroelastic diffusion
constant and elasticity strongly depended on actomyosin (Figure 5-9). The experiments
142
qualitatively illustrate the relative importance of x and E in determining pD .
Depolymerising the F-actin cytoskeleton decreased E and increased pore size resulting
in an overall increase in pD . Conversely, when actin was ectopically polymerised in the
cytoplasm by arp2/3 activation by CA-WASp (Figure 5-8), E increased and the pore size
decreased resulting in a decrease in pD . For both perturbations, changes in pore size x
dominated over changes in cellular elasticity in setting pD . For dense crosslinked F-actin
gels, theoretical relationships between the entanglement length l and the elasticity E
suggest that 2 5~ /( )BE k Tk l with k the bending rigidity of the average F-actin bundle
of diameter b , Bk the Boltzmann constant, and T the temperature [84] (Figure 5-12A).
If the hydraulic pore size x and the cytoskeletal entanglement length l were identical,
pD would scale as 2 3~ /( )p BD k Tk m l implying that changes in elastic modulus would
dominate over changes in pore size, in direct contradiction with our results. Hence, x and
l are different and x may be influenced both by the cytoskeleton and macromolecular
crowding [175] (Figure 5-12A).
To decouple changes in elasticity from gross changes in intracellular F-actin
concentration, we decreased E by reducing F-actin crosslinking through overexpression
of a mutant α-actinin [172] that can either increase the entanglement length l or decrease
the bending rigidity k of F-actin bundles by diminishing their average diameter b
(Figure 5-12). Overexpression of mutant α-actinin led to a decrease in E but no
detectable change in pD or x confirming that pore size dominates over elasticity in
determining cell rheology. Finally, myosin inhibition led to an increase in pD , a decrease
in E , and an increase in x , indicating that myosin contractility participates in setting
rheology through application of pre-stress to the cellular F-actin mesh [176], something
that results directly or indirectly in a reduction in pore size (Figure 5-12B, [176, 177]).
Taken together, these results show that F-actin plays a fundamental role in modulating
cellular rheology but further work will be necessary to understand the relationship
143
between hydraulic pore size x , cytoskeletal entanglement length l , crosslinking, and
contractility in living cells.
Figure 5-11 Effect of volume changes, chemical and genetic treatments on the lumped pore size.
Effect of volume changes, on lumped pore size of HeLa cells in (A) and MDCK cells in (B). (C) Effect of
drug treatments and genetic perturbations on the lumped pore size of Hela cells. The lumped pore size was
estimated from the ratio 1/2( / )pD Em . Asterisks indicate significant changes ( p < 0.01). N is the total number
of measurements and n the number of cells examined.
144
Recent experiments using magnetic bead twisting rheometry have demonstrated the
existence of at least two regimes in the rheology of living cells [80, 178] and in vitro
cytoskeletal networks of actomyosin [176]. At high frequencies (5 Hz - 10 kHz), the
cellular dynamic modulus increased with frequency; whereas at lower frequencies (0.1-
5Hz) it appeared to be frequency independent. At high frequencies (~ 1 kHz ), fast
thermal fluctuations give rise to bending of cytoskeletal filaments and determine the
cellular mechanical properties [80]. However, at longer time-scales (0.2 s to hundreds of
s) cytoskeletal turnover or water movement may influence cell rheology. Measurements
of cell rheology obtained using different techniques over time-scales of 1-100 s reveal the
existence of different regimes: one with a short time-scale (~ 1 s) and the other with a
long time-scale (~ 10 s) [51, 52, 68]. Over the time-scales of the experiments described in
this thesis (~ 5 s), other factors such as turnover of cytoskeletal fibres and cytoskeletal
network rearrangements due to crosslinker exchange or myosin contractility might also in
principle influence cell rheology. In the cells studied here, F-actin, the main cytoskeletal
determinant of cellular rheology (Figure 5-9 and [62, 179]), turned over with a half-time
of ~ 11 s (Figure 5-4B), crosslinkers turned over in ~ 20 s [180], and myosin inhibition
led to faster force-relaxation. Hence, active biological remodelling cannot account for the
dissipative effects observed in our force-relaxation experiments. Taken together, the time-
scale of force-relaxation, the functional form of force-relaxation, and the qualitative
agreement between the theoretical scaling of pD with experimental changes to E and x
support our hypothesis that water redistribution is the principal source of dissipation at
short time-scales contributing ~ 50% of total relaxation in ~ 0.2 s in our experiments.
145
In both viscoelastic and poroelastic models, force decays exponentially with time during
stress-relaxation (see equations (3.5) and (3.6)). Hence, equating the exponents and
recalling 1 ~k E , I obtain the following relationship for a length-scale dependent
effective cellular viscosity h :
2
~L
h mx
(5.1)
with L the characteristic length-scale and m the viscosity of cytosol. The dependence of
η on the ratio of a mesoscopic length-scale to a microscopic length-scale in the system
may explain the large spread in reported measurements of cytoplasmic viscosities [58, 68]
(ranging from ~ 0.1 to 500 pa.s using different rheological measurements such as
magnetic twisting cytometry [33] and AFM indentation [51, 54] experiments).
Within the poroelastic regime, further intuition for the complexity and variety of length
scales involved in setting cellular rheology (Figure 5-12A) can be gained by recognising
that the effective viscosity m experienced by a particle diffusing in the cytosol will
depend on its size (Figure 5-5, [70, 72]). Within the cellular fluid fraction, there exists a
wide distribution of particle sizes with a lower limit on the radius set by the radius of
water molecules. Whereas measuring the poroelastic diffusion constant pD remains
challenging, the diffusivity mD of any given particle can be measured accurately in cells
with experimental techniques such as the presented FRAP technique. For a molecule of
radius a , the Stokes-Einstein relationship gives /(6 )m BD k T apm or
( ) /(6 )B ma k T D am p . Any interaction between the molecule and its environment (e.g.
reaction with other molecules, crowding and hydrodynamic interactions [181], size-
exclusion [70]) will result in a deviation of the experimentally determined mD from this
relationship. Using the relationship for elasticity of gels 2 5~ /( )BE k Tk l and recalling
that the bending rigidity of filaments scales as 4~ polymerE bk (with b the average
146
diameter of a filament –or bundle of filaments-, and polymerE the elasticity of the
polymeric material [182]), yields the relationship
22 8
2 5~
( )polymer
p mB
aE bD D
k T
xl
(5.2)
in which four different length-scales contribute to setting cellular rheology. We see that
the average filament bundle diameter b , the size of the largest particles in the cytosol a
(and indeed the particle size distribution in the cytosol), the entanglement length l , and
the hydraulic pore size b conspire to determine the geometric, transport, and rheological
complexity of the cell (Figure 5-12A). Since all these parameters can be dynamically
controlled by the cell, it is perhaps not surprising that a rich range of rheologies has been
experimentally observed in cells [32, 33, 56, 58, 63-65, 68, 79, 80].
147
Figure 5-12 Schematic representation of the cytoplasm and effects of different cellular perturbations.
(A) The cytoskeleton and macromolecular crowding participate in setting the hydraulic pore size through
which water can diffuse. The length-scales involved in setting cellular rheology are the average filament
diameter b , the size a of particles in the cytosol, the hydraulic pore size x , and the entanglement length l
of the cytoskeleton. (B) Effects of different perturbations on cell rheology. (I, III) Reduction in cell volume
(I) or increase in F-actin concentration (III) causes a decrease in cytoskeletal mesh size l and an increase in
crowding which combined lead to a decrease in hydraulic pore size x . (II, IV) Increase in activity of myosin
motors or crosslinkers lead to either contraction of F-actin network (decrease in mesh size) or increase in
bending rigidity of actin cytoskeleton, which combined results in an increase in the elasticity E of the
network and a decrease in the hydraulic pore size x .
148
Chapter 6
General discussion and future
work
6.1 Further discussion
6.1.1 Cell rheology and its complexity
Living cells are complex materials displaying a high degree of structural hierarchy and
heterogeneity coupled with active biochemical processes that constantly remodel their
internal structure. Therefore, perhaps unsurprisingly, they display an astonishing variety
of rheological behaviours depending on amplitude, frequency and spatial location of
loading. Over the years, a rich phenomenology of rheological behaviours has been
uncovered in cells such as scale-free power law rheology, strain stiffening, anomalous
diffusion, and rejuvenation (see Chapter 1.4.3 and [56, 58, 63-65]).
Several theoretical models have been proposed to explain the observed behaviours but
finding a unifying theory has been difficult because different microrheological
149
measurement techniques excite different modes of relaxation [58]. These rheological
models range from those that treat the cytoplasm as a single phase material whose
rheology is described using networks of spring and dashpots [68, 79], to the sophisticated
soft glassy rheology (SGR) models that describe cells as being akin to soft glassy
materials close to a glass transition [33, 80]; in either case, the underlying geometrical
and biophysical phenomena remain poorly defined [58, 63, 64]. Furthermore, none of the
models proposed, account for dilatational changes in the multiphase material that is the
cytoplasm; these volumetric deformations are ubiquitous in the context of phenomena
such as blebbing, cell oscillations or cell movement [100, 150], and in gels of purified
cytoskeletal proteins [32] whose macroscopic rheological properties depend on the gel
structural parameters and its interaction with an interstitial fluid [81, 84, 176]. Any
unified theoretical framework that aims to capture the rheological behaviours of cells and
links these behaviours to cellular structural and biological parameters must account for
both the shear and dilatational effects seen in cell mechanics as well as account for the
role of crowding and active processes in cells.
6.1.2 Why poroelasticity?
The flow of water plays a critical part in such processes. Recent experiments suggest that
pressure equilibrates slowly within cells (~10s) [100, 101, 103] giving rise to intracellular
flows of cytosol [150, 183] that cells may exploit to create lamellipodial protrusions [150]
or blebs [100, 102] for locomotion [184]. Furthermore, the resistance to water flow
through the soft porous structure serves to slow motion in a simple and ubiquitous way
that does not depend on the details of structural viscous dissipation in the cytoplasmic
network. Based on these observations, a coarse-grained biphasic description of the
cytoplasm as a porous elastic solid meshwork bathed in an interstitial fluid (e.g.
poroelasticity [116] or the two-fluid model [185]) has been proposed as a minimal
framework for capturing the essence of cytoplasmic rheology [59, 100, 101]. In the
framework of poroelasticity, coarse graining of the physical parameters dictating cellular
150
rheology accounts for the effects of the interstitial fluid and related volume changes,
macromolecular crowding and a cytoskeletal network [100, 101, 105], consistent with the
rheological properties of the cell on the time-scales needed for redistribution of
intracellular fluids in response to localised deformation. The response of cells to
deformation then depends only on the poroelastic diffusion constant pD , with larger
values corresponding to more rapid stress relaxations. This single parameter scales as
2~ /pD Ex m , allowing changes in cellular rheology to be predicted in response to
changes in E , x , and m .
The principal shortcoming of most models of cell rheology is that they do not relate the
measured rheological properties to structural or biological parameters within the cell
meaning that they lack predictive power. As a coarse-grained bottom-up approach,
one advantage of using poroelasticity over other phenomenological models is that it
allows cellular rheology to be related to observable physical and biological parameters,
making it mechanistic and thus giving it predictive power (see Figure 6-1). In this work, I
sought to understand cell rheology at physiologically relevant time-scales and to attempt
to understand the cellular biological and physical variables that affect it. By nature,
cellular rheology is bound to be extremely complex but my goal was to capture
its essence with as few physical parameters as possible. The poroelastic model of the
cytoplasm presented here is a first step in this direction. It takes as inputs
measurable geometric and physical parameters and leads to falsifiable predictions (see
Figure 6-1). This represents a conceptual advance and a step towards describing the
experimental observations with the fewest physiologically and biologically relevant
parameters possible.
151
Figure 6-1 Microstructural parameters and mesoscopic length scale set the poroelastic time scale.
Different timescales associated with different rheological models are involved in setting cell rheology. The
time-dependent mechanics of cells arise from the combination of several distinct rheological behaviours.
These rheological behaviours give rise to different dynamics that coexist but each dominates over a range of
timescales. At very short time scales, the behaviour of network of semiflexible filaments (entropic and
enthalpic fluctuations) sets the rheology of the cytoplasm [80]. At long timescales, the cytoplasm exhibits
inelastic structural rearrangements and behaves as a soft glassy material [80]. At very long timescales the cell
can no longer be considered as a passive material and active processes such as molecular motor activities and
cytoskeletal filament turnover come into play. Depending on the length scale and time scale, cytosolic flows
and intracellular fluid redistribution (poroelastic effects) can affect the rheological behaviour of the cell. The
mesoscopic length scale of the system L , the elastic modulus of cell E , the hydraulic pore size of the
cytoplasm x , and the cytosolic viscosity m set the characteristic poroelastic timescale pt ( ~ 0.2pt s in my
AFM indentation experiments).
Time dependent mechanics of cells is the manifestation of several cell rheological
behaviours and these rheological behaviours give rise to different dynamics that could
coexist but each dominates on different timescales. In particular, poroelastic cellular
behaviours, in which dissipations are due to water redistribution, are both timescale and
lengthscale dependent allowing us to tune the experimental setup in such a way that we
observe weak or strong poroelastic effects accordingly. The idea of designing this study
was to consider a minimal poroelastic model under optimised experimental conditions,
and determine whether this minimal model was sufficient to grasp the essence of cellular
rheology and predict changes in rheology in response to perturbations. Extra complexity
could be added to the model to take into account more complex properties of the
cytoskeleton and cytoplasm or network tension: we could consider a solid phase with
152
intrinsic viscoelasticity and/or granularity instead of a simple linear elastic meshwork.
However, this would come at the expense of introducing more parameters and extra
uncertainty as one would have to make many more assumptions about cell mechanics and
the end result would not necessarily draw a clearer description of cell rheology and its
origins. In particular, because the exact relationship between fluid fraction, volumetric
pore size , hydraulic pore size , and cytoskeletal entanglement length is still
unknown for living cells, rather than making complex hypotheses about the poroelastic
scaling law, the trends of my experimental results were validated.
Future studies should concentrate on implementing more realistic cytoskeletal and
cytoplasmic properties. However addition of extra complexity should be coupled with
experimental determination/validation of all newly introduced variables and this might
require much more sophisticated experiments.
6.1.3 Direct physiological relevance
To gain an understanding of how widespread poroelastic effects are in the rheology of
isolated cells and cells within tissues, we can compute the poroelastic Péclet number
eP ( )/ pVL D , with V a characteristic velocity (due to active movement, external
loading, etc). For eP large compared to 1, poroelastic effects dominate the viscoelastic
response of the cytoplasm to shape change due to externally applied loading or intrinsic
cellular forces. In isolated cells, poroelastic effects have been implicated in the formation
of protrusions such as lamellipodia or blebs [100]. For these, the rate of protrusion growth
can be chosen as a characteristic velocity. In rapidly moving cells, forward-directed
intracellular water flows [150, 183] resulting from pressure gradients due to myosin
contraction of the cell rear have been proposed to participate in lamellipodial protrusion
[150, 186]. Assuming a representative lamellipodium length of L ~10 µm and protrusion
velocities of V ~ 0.3 µm.s-1
, poroelastic effects will play an important role if pD ≤ 3
µm2.s
-1, lower than measured in the cytoplasm but consistent with the far higher F-actin
153
density observed in electron micrographs of the lamellipodium [184]. Furthermore, cells
can also migrate using blebbing motility [107] where large quasi-spherical blebs (L
~10µm) arise at the cell front with protrusion rates of V ~ 1 µm.s-1
giving an estimate of
pD ~ 10 µm
2.s
-1 (comparable to the values reported in previous Chapter ) to obtain eP
≥
1.
During normal physiological function, cells within tissues of the respiratory and
cardiovascular systems are subjected to strains e > 20% applied at strain rates te > 20%.
s-1
. As a first approximation, we assume that these cells, with a representative length cellL ,
undergo a length change cell~L Le applied with a characteristic velocity cell~ tV Le . For
cells within the lung alveola [155], cellL ~ 30 µm, e ~ 20%, te
~ 20%. s-1
and assuming
pD ~ 10 µm
2.s
-1 (based on our measurements), we find eP
~ 3. Hence, these simple
estimates of eP suggest that water redistribution participates in setting the rheology of
cells within tissues under normal physiological conditions.
6.2 Future work
6.2.1 Stress and pressure wave propagation in cells
It is well established that cells can sense mechanical forces, react to them, and adapt to
them [58]. However, how these forces are sensed and in particular what physical variable
is detected by cells remains unclear. A better understanding of cell rheology will allow us
to predict the intracellular stress distribution induced by extracellular application of force.
154
Figure 6-2 Stress and pressure wave propagation in cells
Application of step force/pressure via AFM cantilever (A) or through micropipette (B). Upon application of
force/pressure the stress and pressure propagate intracellularly. (C-I) The displacement of each fluorescent
bead in the z direction is monitored as a function of time and the bead distance from the force/pressure
application point is measured. The bead displacement relaxation time increases with distance from the source.
(C-II) Expected curve for the relaxation time as a function of distance. This curve can provide information
about how fast stress/pressure propagates through the cytoplasm. Comparing these experimental data with
viscoelastic and poroelastic models will provide information about the physical nature of force transmission
inside the cytoplasm.
In the future, matching the temporal evolution of this intracellular stress distribution with
biological readouts (such as the intracellular calcium concentration or the turnover rates
of cytoskeletal proteins) should allow us to better understand what physical variable cells
sense and how it is sensed at the molecular level prior to conversion into a biochemical
signal. In particular, recent experiments suggest that poroelastic effects lead to slow stress
propagation within living cells [103]. In an attempt to understand how stresses propagate
inside the cytoplasm, future work will try to apply localized force or pressure onto the
cell via an AFM cantilever or through a micropipette (see Figure 6-2A, B). The particle
155
tracking technique as described in Chapter 3.10 will be used to monitor motion of
attached beads at different distances from the point of exerted force/pressure. Studying
the relaxation curves as a function of distance (see Figure 6-2C) will provide information
about how fast stress/pressure propagates through the cytoplasm. Comparing these
experimental data with viscoelastic and poroelastic models will provide useful
information about which physical mechanism are effectively involved in force
transmission inside the cytoplasm.
6.2.2 Cells as multi-layered poroelastic materials
From a hydrodynamic perspective, the cell can be viewed as a multilayer composite
composed of plasma membrane attached to the subcortical actin gel surrounding
a molecularly crowded environment saturated with cytosol. One future challenge will be
to dissect the contribution of each layer of the cell to cell rheology. In Chapter 5, AFM
indentation tests identified the actin cytoskeleton as the main component of the cytoplasm
that affected the poroelastic rheology of the cell. It would be interesting to investigate the
contribution of the different layers of the cell to the swelling/shrinking experiments
presented in Chapter 4.2. In particular, my preliminary experimental results (not
presented here) suggested that the plasma membrane can play a significant role in setting
the dynamics of cell swelling/shrinking.
I propose to use defocusing microscopy to study the kinetics of cell swelling/shrinking
under different types of perturbations to investigate the multi-layered poroelastic nature
of the cell and to see if the contribution of each layer in cell rheology can be dissected.
First, to examine the role of membrane permeability, cells can be treated with small
concentration of mild detergents or pore forming proteins such as Digitonin or
Amphotericin B to make small holes in the membrane. It should be noted that after
membrane permeabilisation the cell will start swelling. As examined in Chapter 5,
swelling can affect the poroelastic properties (specifically pore size) significantly and
therefore to counterbalance the effects of cell swelling upon membrane permeabilisation,
156
a small concentration of osmolyte such as sucrose must be added to retain the isoosmotic
cell volume. I would expect the poroelastic diffusion coefficient to increase when the
membrane is permeabilised but the extent to which it increases will tell us about the
importance of membrane. Next, the effects of macromolecular crowding can be envisaged
by using higher concentrations of detergents to create holes large enough for
macromolecules to escape (up to micrometres in size). It should be noted that creation of
large holes in membrane might result in clearance of some of the basic components of the
cytoskeleton (such as actin monomers) that will affect the structure of filaments and
consequently the the cellular poroelastic properties.
Second, I will investigate the contribution of the actin cytosleketon to swelling/shrinking
dynamics by using drugs such as latrunculin B to depolymerise the actin filaments or
performing genetic treatments (as explained in Chapter 3.2) to induce excessive
polymerization of F-actin in the cytoplasm. Treatment of cells with drugs such as
latrunculin B will affect the structure of the whole actin cytoskeleton including the
organization of cortical actin. Therefore, to study the role of cell cortex in setting cell
rheology and poroelastic properties separately and in particular to solely perturb structure
and organization of the cortical actin, it would be very interesting to conduct more
targeted perturbations such as shRNA gene depletion to knock down the protein Diaph1
(a protein shown by other members of the laboratory to be responsible for polymerisation
of actin filaments in the cortex) or using the drug CK666 to inhibit the Arp2/3 complex
that alsoregulates the structure of the actin network.
157
6.3 Conclusions
The main conclusions arising from this work may be summarised as follows:
- The time-dependent mechanics of living cells exhibit poroelastic behaviours similar
to poroelastic physical gels.
- The dynamics of cell swelling and shrinking can be well described under the
framework of poroelasticity.
- Poroelasticity predicts changes in rheology due to cell volume changes and
cytoskeletal disruption.
- F-actin and molecular crowding are the main cytoskeletal contributors of poroelastic
cellular rheology and together they set the rheological properties of the cell.
- In my experiments, the observed short timescale (0.1-1 s) cellular viscosity is due to
water redistribution in the cell.
158
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